WEBVTT
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YUFEI ZHAO: OK,
let's, get started.
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Welcome to 18.217.
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So this is combinatorial
theory, graph theory,
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and additive combinatorics.
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So course website is up there.
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So all the course
information is on there.
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So after around the
middle of the class,
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I'll say a bit more about
various course information,
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administrative things.
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But I want to jump directly
into the mathematical content.
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So this course
roughly has two parts.
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The first part will
look at graph theory,
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in particular problems
in extremal graph theory.
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In the second part,
we'll transition
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to additive combinatorics.
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But these are not two
separate subjects.
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So I want to show you
this topic in a way that
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connects these two
areas and show you
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that they are quite
related to each other.
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And many of the
common themes that
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will come up in one
part of the course
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will also show up in the other.
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So the story between
graph theory and additive
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combinatorics began
about 100 years ago
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with Schur, the famous
mathematician, Isaai Schur.
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Well, he was like
many mathematicians
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of his era trying to prove
Fermat's Last Theorem.
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So here's what's
Schur's approach.
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He said, well, let's look at
this equation that comes up
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in from Fermat's Last Theorem.
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And, well, one of the methods
of elementary number theory
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to rule out solutions
to an equation
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is to consider what
happens when you mod p.
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If you can rule out for
infinitely many values
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p, possible non-trivial
solutions to this equation mod
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p, then you will rule out
possibilities of solutions
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to Fermat's Last Theorem.
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OK, so this was
Schur's approach.
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As you can guess, unfortunately,
this approach did not work.
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And Schur proved that this
method definitely doesn't work.
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So that's the starting
point of our discussion.
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So it turns out that
for every value of n,
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there exists
non-trivial solutions
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for all p sufficiently large.
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So thereby, ruling
out the strategy.
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So let's see how Schur
proved his theorem.
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So that will be the first
half of today's lecture.
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So this seems like a
number theory question.
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So what does it have to
do with graph theory?
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So I wanted to show
you this connection.
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Now, Schur deduced his
theorem from another result.
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That is known as
Schur's Theorem, which
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says that if be
positive integers
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is colored using
finitely many colors,
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then there exists a
monochromatic solution
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to the equation x
plus y equals to z.
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So if you give me
10 colors and color
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the positive integers
using those 10 colors,
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then I can find
for you a solution
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to this equation where x, y,
and z are all of the same color.
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Now, this statement--
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OK, so it's a perfectly
understandable statement.
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But let me rephrase it in
a somewhat different way.
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And this gets to a
point that I want
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to discuss where many statements
in additive combinatorics
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or just combinatorics in general
have different formulations,
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one that comes in an
infinitary form, which
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is more qualitative so to
speak and another form that
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is known as finitary.
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And that's more
quantitative in nature.
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So Schur's Theorem is
stated in a infinitary form.
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So it tells you if you color
using finitely many colors,
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then there exists a
monochromatic solution.
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So many, but not all,
statements of that form
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have an equivalent finitary form
that is sometimes more useful.
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And also, once you stay
the right finitary form,
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you can ask
additional questions.
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So here's what Schur's Theorem
looks like in the equivalent
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finitary form.
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You give me an r.
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For every r, there exists
some N as a function of r,
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such that if the
numbers 1 through N--
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so throughout this course, I'm
going to use this bracket N
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to denote integers up to N--
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so if these numbers are
colored using our colors,
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then necessarily, there exists
a monochromatic solution
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to the equation x
plus y equals to z,
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where x, y, and z are in the
set that is being colored.
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So it looks very similar to
the first version I stated.
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But now, there are
some more quantifiers.
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So for every r,
there exists an N.
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So why are these two versions
equivalent to each other?
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So it's not too hard to
deduce their equivalence.
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So let me do that now.
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The fact that the
finitary version
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implies the infinitary
version claims
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should be fairly obvious.
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So once you know the
finitary version,
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if you give me a coloring of
the positive integers, well
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I just have to look
far enough up to this N
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and I get the conclusion I want.
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But now, in the
other direction--
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so in the other direction--
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suppose I fix to this r.
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So, OK, so I assume
the infinitary version.
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I wanted to deduce
the finitary version.
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So I start with this r.
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And let's suppose the
conclusion were false.
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So supposed the
conclusion were false,
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namely for every N there
exists some coloring--
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so for every N there
exists some coloring--
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which we will call
phi sub N, that
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avoids monochromatic solutions
to x plus y equals to z.
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So I'm going to use this Chi
for shorthand for monochromatic.
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So suppose there
exists such a coloring.
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And now, I want to take
this collection of colorings
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and produce for you a coloring
of the positive integers.
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And you can do this basically
by a standard diagonalization
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trick.
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Namely, we see that by taking
an infinite subsequent,
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such that--
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so let me call this infinite
sub-sequence phi of--
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phi sub-- well, so it's infinite
sub-sequence of this phi sub N,
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such that phi sub N
of k stabilizes along
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the sub-sequence for every k.
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OK, so you can do this simply
by diagonalization trick.
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And then, we see that
along the sub-sequence phi
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N converges point-wise to some
coloring of the entire set
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of positive integers.
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And this coloring avoids
monochromatic solutions
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to x plus y equals to
z, because if there
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were monochromatic
solutions in this coloring
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of the entire integers,
then I can look back
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to where that came from.
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And that would have been the
monochromatic solution in one
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of my phi N's.
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So this is an
argument that shows
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the equivalence between the
finitary form and infinitary
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form.
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But now, when we look
at the finitary form,
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you can ask additional
questions, such as,
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how big does this N have
to be as a function of r.
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It turns out those kind
of questions in general
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are very difficult. And
we know some things.
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For this type of questions,
we know some bounds usually.
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But the truth is
usually unknown.
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And there are
major open problems
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in combinatorics of this type.
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So there's still a lot
that we do not understand.
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OK, so now we have Schur's
Theorem in this form.
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Let me show you how to deduce
his conclusion about ruling out
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this approach to proving
Fermat's Last theorem.
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The claim is the following that
if you have a positive integer
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n, then for all
sufficiently large primes p,
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there exists x, y,
and z, all belonging
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to integers up to p minus 1,
such that their n-th powers
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add up like this.
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So it's a solution to
Fermat's equation mod p.
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All right, so how
can we deduce this
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from what we said
about coloring?
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So what is the coloring?
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OK, so here's what Schur
did, so proof assuming
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for now Schur's theorem.
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So let's look at the
multiplicative group
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of non-zero residues, mod p.
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So we know it's a cyclic group
because there's a generator.
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So there's a primitive
root generator.
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Let H denote the
subgroup of n-th powers.
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Well, H is a pretty
big subgroup.
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So what's the index of H in
this multiplicative group?
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It's at most M.
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So think about representing
this as a cyclic group using
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a generator.
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So H then would be all the
elements whose exponent
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is divisible by M. So
this the index is at most
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M. It could be smaller.
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But it's at most M.
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And so in particular,
I can use the H cosets
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to partition the multiplicative
group of non-zero residues.
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And this is a color.
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Virtual partition is the
same thing as a coloring.
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There is a bounded
number of colors.
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But I let peek at large.
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So by Schur's theorem if
p is sufficiently large,
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then one of my cosets
should contain a solution
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to x plus y equals to z.
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What does that look like?
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So that one coset, one H coset,
course that contains x, y, z,
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such that x plus y
equals to z as integers.
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They belong to the same coset.
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So x, y, and z belong
to some coset of H,
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which means then that x equals
to a times n-th power with a y
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equals to a times some
n-th power and little z
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equals to a times
some n-th power.
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You have this equation.
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Put them together.
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So that is true.
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So now mod p, I
can cancel the a's.
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And this produces a
non-trivial solution
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to Fermat's equation, mod p.
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OK, so this was the
proof of this claim
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that this method does
not work for solving
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Fermat's Last Theorem.
00:16:34.160 --> 00:16:38.190
But, you know, we assumed
this claim of Schur's theorem
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that every finite coloring
of the positive integers
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contains a monochromatic
solution to x plus y
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equals to z.
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So we still need to
prove that claim.
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So we still need to prove
this combinatorial claim.
00:16:49.302 --> 00:16:51.010
And so that's what
we're going to do now.
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This is where graph
theory comes in.
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So let me state a very
similar-looking theorem
00:17:23.359 --> 00:17:25.280
about graphs.
00:17:25.280 --> 00:17:27.793
And this is known
as Ramsey's theorem,
00:17:27.793 --> 00:17:29.960
although Ramsay's theorem
actually historically came
00:17:29.960 --> 00:17:33.590
after Schur's theorem, but
Ramsey's theorem, here, we're
00:17:33.590 --> 00:17:36.230
going to use it specifically
in the case for triangles.
00:17:44.680 --> 00:17:45.620
So what does it say?
00:17:45.620 --> 00:17:50.460
That if you give me an
r, the number of colors,
00:17:50.460 --> 00:17:55.920
then there exists
some large N such
00:17:55.920 --> 00:18:13.043
that if the edges of the
complete graph, K sub N,
00:18:13.043 --> 00:18:26.870
along N vertices are
colored using r colors,
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then there exists a
monochromatic triangle
00:18:32.990 --> 00:18:33.907
somewhere.
00:18:45.850 --> 00:18:48.371
Any questions so far about
any of these statements?
00:18:51.530 --> 00:18:54.260
So let's see how Ramsay's
theorem for triangles
00:18:54.260 --> 00:18:54.760
is proved.
00:19:08.650 --> 00:19:11.890
By the way, I want to give you
a historical note about Frank
00:19:11.890 --> 00:19:13.010
Ramsey.
00:19:13.010 --> 00:19:17.830
So he's someone who made
significant contributions
00:19:17.830 --> 00:19:20.770
to many different areas,
not just in mathematics.
00:19:20.770 --> 00:19:24.010
So he contributed to seminal
works in mathematical logic
00:19:24.010 --> 00:19:27.280
where this theorem came
from, but also to philosophy
00:19:27.280 --> 00:19:32.230
and to economics before his
untimely death at the age of 26
00:19:32.230 --> 00:19:35.520
from liver-related problems.
00:19:35.520 --> 00:19:39.670
So he's someone whose very short
life contributed tremendously
00:19:39.670 --> 00:19:42.870
to academics.
00:19:42.870 --> 00:19:45.590
So let's see how Ramsay's
theorem, in this case,
00:19:45.590 --> 00:19:47.780
is proved.
00:19:47.780 --> 00:19:52.830
We'll do induction on
r, the number of colors.
00:19:56.740 --> 00:19:59.580
So for every r, I need
to show you some N,
00:19:59.580 --> 00:20:01.430
such that the statement is true.
00:20:01.430 --> 00:20:07.040
In the first case, when r equals
to 1, there's not much to do.
00:20:07.040 --> 00:20:10.080
Just one color, if I
just have three vertices,
00:20:10.080 --> 00:20:12.830
that already is OK.
00:20:12.830 --> 00:20:16.220
Three vertices, that's already
a monochromatic triangle.
00:20:16.220 --> 00:20:21.390
So from now on, let
r be at least 2.
00:20:21.390 --> 00:20:35.480
And suppose the claim
holds for r minus 1 colors,
00:20:35.480 --> 00:20:42.440
with N prime being the
corresponding number
00:20:42.440 --> 00:20:46.280
of vertices with
r minus 1 colors.
00:20:46.280 --> 00:20:48.185
So now let me pick
an arbitrary vertex.
00:20:53.260 --> 00:20:57.290
So pick an arbitrary vertex
v and look what happens.
00:20:57.290 --> 00:20:59.810
So here's v.
00:20:59.810 --> 00:21:03.620
And let me look at
the outgoing edges.
00:21:09.940 --> 00:21:16.930
So we'll show that
N being r crimes
00:21:16.930 --> 00:21:21.650
N prime minus 1 plus 2 works.
00:21:21.650 --> 00:21:26.260
So now, we have a lot
of outgoing edges.
00:21:26.260 --> 00:21:31.120
In particular, we
have r times N prime
00:21:31.120 --> 00:21:35.530
minus 1 plus 1 outgoing edges.
00:21:35.530 --> 00:21:45.820
So by the pigeonhole
principle, some color--
00:21:45.820 --> 00:21:56.180
so there exists at least
N prime outgoing edges
00:21:56.180 --> 00:22:05.110
with the same color,
let's say, yellow.
00:22:05.110 --> 00:22:07.370
So suppose yellow is
the outgoing color.
00:22:14.260 --> 00:22:16.420
And let me call
the set of vertices
00:22:16.420 --> 00:22:18.460
on the other end
of these edges v0.
00:22:22.520 --> 00:22:24.736
So now let's think about
what happens in v0.
00:22:24.736 --> 00:22:41.020
So in v0, either v0 contains
a yellow edge, in which case
00:22:41.020 --> 00:22:42.880
you get a yellow triangle.
00:22:53.002 --> 00:22:56.163
Or we lose the color inside v0.
00:22:56.163 --> 00:22:57.580
So the number of
colors goes down.
00:23:00.835 --> 00:23:11.680
Else v0 has at most
r minus 1 colors.
00:23:11.680 --> 00:23:22.090
And v0 has at least N
prime number of vertices.
00:23:22.090 --> 00:23:32.600
So by induction, v0 has
a monochromatic triangle
00:23:32.600 --> 00:23:34.030
in the remaining colors.
00:23:40.180 --> 00:23:42.880
So that completes the proof of
Ramsay's theorem, in this case,
00:23:42.880 --> 00:23:44.860
for triangles.
00:23:44.860 --> 00:23:49.337
And if you wish to find out
what is the bound that comes out
00:23:49.337 --> 00:23:51.420
of this argument, well,
you can chase to the proof
00:23:51.420 --> 00:23:52.212
and get some bound.
00:23:55.098 --> 00:23:57.140
The remaining question
now is, what does this all
00:23:57.140 --> 00:23:59.990
have to do with Schur's theorem?
00:23:59.990 --> 00:24:02.760
So so far, we've talked
about some number theory.
00:24:02.760 --> 00:24:04.320
We've talked about
some graph theory
00:24:04.320 --> 00:24:07.350
and how to link these
two things together.
00:24:07.350 --> 00:24:08.850
And I think this
is a great example.
00:24:08.850 --> 00:24:10.520
It's a fairly simple
example, which
00:24:10.520 --> 00:24:14.240
I'm about to show you of how to
link these two ideas together.
00:24:14.240 --> 00:24:16.985
And this connection,
we'll see many times
00:24:16.985 --> 00:24:18.110
in the rest of this course.
00:24:28.820 --> 00:24:30.650
I don't want to erase
Schur's theorem.
00:24:30.650 --> 00:24:31.390
So let me--
00:24:47.570 --> 00:24:50.510
So let's prove Schur's theorem.
00:24:57.820 --> 00:24:59.190
So let's start with a coloring.
00:25:10.174 --> 00:25:14.130
So let's start with the
coloring of 1 through N.
00:25:14.130 --> 00:25:18.510
And I want to form
a graph with colors
00:25:18.510 --> 00:25:21.090
on the edges that
are somehow derived
00:25:21.090 --> 00:25:24.210
from this coloring
on these integers.
00:25:24.210 --> 00:25:26.070
And here's what I'm going to do.
00:25:26.070 --> 00:25:30.110
So let's color the
complete graph-- let's
00:25:30.110 --> 00:25:46.320
color the edges of the complete
graph on the vertex set
00:25:46.320 --> 00:25:49.470
having N plus 1 vertices,
labeled at integers
00:25:49.470 --> 00:25:54.160
up to positive integers
up to N plus 1.
00:25:54.160 --> 00:26:08.060
But by the Ramsey result we just
proved, if N is large enough,
00:26:08.060 --> 00:26:09.910
then there exists a
monochromatic triangle.
00:26:21.867 --> 00:26:22.950
So what does it look like?
00:26:26.010 --> 00:26:30.900
So let me draw for you a
monochromatic triangle.
00:26:30.900 --> 00:26:35.280
Suppose it-- so I haven't told
you what the coloring is yet.
00:26:35.280 --> 00:26:37.290
So the coloring
is that I'm going
00:26:37.290 --> 00:26:43.090
to color the edge
between i and j,
00:26:43.090 --> 00:26:48.460
using the color derived by
applying phi to the number
00:26:48.460 --> 00:26:52.150
j minus i, namely the
length of that segment
00:26:52.150 --> 00:26:56.800
if I lay out all the
vertices on the number line.
00:26:56.800 --> 00:27:00.550
So now have an r coloring
of this complete graph.
00:27:00.550 --> 00:27:03.010
So Ramsey tells us
that there exists
00:27:03.010 --> 00:27:05.260
a monochromatic triangle.
00:27:05.260 --> 00:27:10.510
The triangle sits on
vertices i, j, and k.
00:27:10.510 --> 00:27:18.670
And the rule tells us that the
colors are phi of k minus i,
00:27:18.670 --> 00:27:23.470
phi of j minus i,
and phi of k minus j.
00:27:30.810 --> 00:27:33.660
So these three numbers,
they have the same coloring.
00:27:33.660 --> 00:27:41.060
But, look, if I set these
numbers to be x, y, and z--
00:27:41.060 --> 00:27:44.540
so x being j minus
i, for instance--
00:27:44.540 --> 00:27:49.970
then x plus y equals to z.
00:27:49.970 --> 00:27:53.960
And they all have
the same color.
00:27:57.040 --> 00:28:00.420
So this monochromatic
triangle gives us
00:28:00.420 --> 00:28:03.990
a monochromatic
equation, 2x plus y
00:28:03.990 --> 00:28:11.810
equals to z, thereby concluding
the proof of Schur's theorem.
00:28:11.810 --> 00:28:15.660
OK, so this rounds out the
discussion for now of--
00:28:15.660 --> 00:28:21.090
well, we started with some
statement about number theory.
00:28:21.090 --> 00:28:25.960
And then we took this
detour to graph theory,
00:28:25.960 --> 00:28:29.920
looking at Ramsay's theorems
of monochromatic triangles,
00:28:29.920 --> 00:28:32.140
and then go back
to number theory
00:28:32.140 --> 00:28:36.140
and proved the result
that Schur did.
00:28:36.140 --> 00:28:40.260
So how does go to graphs help?
00:28:40.260 --> 00:28:41.940
So why was this advantageous?
00:28:45.642 --> 00:28:46.600
What do you guys think?
00:28:54.290 --> 00:28:58.530
So I claim that by
going to graphs,
00:28:58.530 --> 00:29:04.710
we added some extra flexibility
to what we can play with.
00:29:04.710 --> 00:29:07.620
For example, we started
out with a problem
00:29:07.620 --> 00:29:12.370
where there were only
N things being colored.
00:29:12.370 --> 00:29:15.670
And then we moved to
graphs where about--
00:29:15.670 --> 00:29:19.510
well, N choose 2 or N squared
objects are being colored.
00:29:19.510 --> 00:29:21.840
And then we did an
induction argument.
00:29:21.840 --> 00:29:25.570
So remember in the proof of
Ramsey's theorem up there,
00:29:25.570 --> 00:29:28.900
there was an induction
argument taking all vertices.
00:29:28.900 --> 00:29:33.340
And that argument doesn't
make that much sense
00:29:33.340 --> 00:29:35.155
if you stayed
within the numbers.
00:29:37.790 --> 00:29:39.620
Somehow moving to
graphs gave you
00:29:39.620 --> 00:29:43.310
that extra flexibility
allow you to do more things.
00:29:43.310 --> 00:29:45.440
And this is one
of the advantages
00:29:45.440 --> 00:29:49.480
of moving from
problem about numbers
00:29:49.480 --> 00:29:51.430
to a problem about graphs.
00:29:51.430 --> 00:29:54.030
And we'll see this
connection later on as well.
00:29:54.030 --> 00:29:55.090
Yeah?
00:29:55.090 --> 00:29:56.590
AUDIENCE: Sort of
related to that.
00:29:56.590 --> 00:30:00.590
Are there better bounds
known for this specific,
00:30:00.590 --> 00:30:04.340
like Schur's result
of that power on e,
00:30:04.340 --> 00:30:06.590
because the N's here
would be pretty bad.
00:30:06.590 --> 00:30:10.317
YUFEI ZHAO: Right, so Ashwan
asked, so what about bounds?
00:30:10.317 --> 00:30:11.650
So what do we know about bounds?
00:30:11.650 --> 00:30:14.298
So I don't know off the
top of my head the answers
00:30:14.298 --> 00:30:15.090
to those questions.
00:30:15.090 --> 00:30:17.940
But in general,
they're quite open.
00:30:17.940 --> 00:30:20.880
So there are exponential gaps
between lower and upper bounds
00:30:20.880 --> 00:30:23.122
on our knowledge of what
is the optimal N you
00:30:23.122 --> 00:30:24.080
can put in the theorem.
00:30:26.640 --> 00:30:28.057
Any more questions?
00:30:31.400 --> 00:30:35.280
All right so, I think this is
a good point for us to-- so
00:30:35.280 --> 00:30:37.770
usually when I give
90-minute lectures,
00:30:37.770 --> 00:30:40.920
I like to take a short
2-minute break in between.
00:30:40.920 --> 00:30:41.880
So I want to do that.
00:30:41.880 --> 00:30:43.350
And then in the
second half, I want
00:30:43.350 --> 00:30:47.940
to take you through a tour
of additive combinatorics.
00:30:47.940 --> 00:30:51.020
So tell you about some of
the modern developments.
00:30:51.020 --> 00:30:54.720
Now, this is an exciting
field where it started out,
00:30:54.720 --> 00:30:58.500
I think, roughly with Schur's
theorem that we just discussed.
00:30:58.500 --> 00:31:00.300
That started about
100 years ago.
00:31:00.300 --> 00:31:03.510
But a lot has taken place
in the past century.
00:31:03.510 --> 00:31:06.720
And there's still a lot of
ongoing exciting research
00:31:06.720 --> 00:31:07.470
developments.
00:31:07.470 --> 00:31:09.280
So in the second
half of this lecture,
00:31:09.280 --> 00:31:13.110
I want to give you a tour
through those developments
00:31:13.110 --> 00:31:14.970
and show you some
of the highlights
00:31:14.970 --> 00:31:16.428
from additive combinatorics.
00:31:16.428 --> 00:31:17.970
So let's take a
quick 2-minute break.
00:31:17.970 --> 00:31:22.506
And feel free to ask
questions in the meantime.
00:31:22.506 --> 00:31:24.280
So another part of
the writing assignment
00:31:24.280 --> 00:31:27.970
in addition to course
notes is a contribution
00:31:27.970 --> 00:31:32.198
to Wikipedia, which is, you
know, nowadays, of course,
00:31:32.198 --> 00:31:34.115
you know, if you hear
some word like Szemeredi
00:31:34.115 --> 00:31:36.280
's regularity lemma
the first thing you do
00:31:36.280 --> 00:31:37.570
is type into Google.
00:31:37.570 --> 00:31:39.760
And more often than not the
first link that comes up
00:31:39.760 --> 00:31:41.610
is Wikipedia.
00:31:41.610 --> 00:31:43.260
And, you know, some
of the articles,
00:31:43.260 --> 00:31:47.940
they are all right, and some of
them are really not all right.
00:31:47.940 --> 00:31:51.480
And it would be fantastic
for future students
00:31:51.480 --> 00:31:56.100
and also for yourselves
if there were better entry
00:31:56.100 --> 00:32:00.540
points to this area by having
higher quality Wikipedia
00:32:00.540 --> 00:32:03.330
articles or articles
that are simply
00:32:03.330 --> 00:32:05.530
missing about specific topics.
00:32:05.530 --> 00:32:07.145
So one of the assignments--
00:32:07.145 --> 00:32:08.520
again, this can
be collaborative.
00:32:08.520 --> 00:32:09.895
So I'll give you
more information
00:32:09.895 --> 00:32:11.010
how to do that later--
00:32:11.010 --> 00:32:14.340
is to contribute to
Wikipedia and roughly
00:32:14.340 --> 00:32:16.500
contribute one high
quality article
00:32:16.500 --> 00:32:18.540
or edit some
existing articles so
00:32:18.540 --> 00:32:21.280
that they become high quality.
00:32:21.280 --> 00:32:21.780
Yep.
00:32:21.780 --> 00:32:24.986
AUDIENCE: Can we
something similar to LMDB
00:32:24.986 --> 00:32:27.880
with creating a website
that has all the information
00:32:27.880 --> 00:32:30.007
needed in combinatorics?
00:32:30.007 --> 00:32:31.590
YUFEI ZHAO: So we
can talk about that.
00:32:31.590 --> 00:32:34.770
So if there are other
ideas about how to do this,
00:32:34.770 --> 00:32:37.363
we can definitely open
the chatting about that.
00:32:37.363 --> 00:32:38.780
So the other thing
is that instead
00:32:38.780 --> 00:32:42.680
of holding the usual office
hours, what I like to do is--
00:32:42.680 --> 00:32:45.170
so this class ends at 4:00 PM.
00:32:45.170 --> 00:32:47.330
So after 4:00, I'll go
up to the Math Common
00:32:47.330 --> 00:32:49.090
Room, which is
just right upstairs
00:32:49.090 --> 00:32:50.360
and hang out there for a bit.
00:32:50.360 --> 00:32:53.340
If you have questions, you
want to chat, come talk to me.
00:32:53.340 --> 00:32:57.050
I'd be happy to chat about
anything related or not related
00:32:57.050 --> 00:32:58.030
to the course.
00:32:58.030 --> 00:33:00.290
And before homeworks
are due, I will
00:33:00.290 --> 00:33:03.890
try to set up some special
office hours for you
00:33:03.890 --> 00:33:06.215
in case you want to ask
about homework problems.
00:33:06.215 --> 00:33:08.090
And if you want to meet
with me individually,
00:33:08.090 --> 00:33:09.970
please just send me an email.
00:33:09.970 --> 00:33:11.720
Oh, one more thing
about the course notes.
00:33:11.720 --> 00:33:16.290
So because I want to
do quality control,
00:33:16.290 --> 00:33:19.910
so here is the process that will
happen with the course notes.
00:33:19.910 --> 00:33:21.890
So the first lecture
is already online.
00:33:21.890 --> 00:33:22.940
So you can already see.
00:33:22.940 --> 00:33:25.790
So I've written up the lecture
notes for the first lecture.
00:33:25.790 --> 00:33:27.620
And you can use
that as an example
00:33:27.620 --> 00:33:30.210
of what I'm looking for.
00:33:30.210 --> 00:33:32.098
So I'm looking for
people to sign up
00:33:32.098 --> 00:33:33.640
starting from the
next lecture, and I
00:33:33.640 --> 00:33:36.470
will send out a link tonight.
00:33:36.470 --> 00:33:41.040
For future lectures, so
whoever writes the lecture,
00:33:41.040 --> 00:33:45.290
I'll the lecture, and
then within one day, so
00:33:45.290 --> 00:33:48.250
by the end of the day
after the lecture,
00:33:48.250 --> 00:33:51.080
it will be good if
they were already
00:33:51.080 --> 00:33:53.065
at least some sketch,
some rough draft at least
00:33:53.065 --> 00:33:54.440
containing the
theorem statements
00:33:54.440 --> 00:33:56.390
and whatnot from
the day's lecture.
00:33:56.390 --> 00:34:01.100
So that the next person can
start writing afterwards.
00:34:01.100 --> 00:34:03.290
But once you are
done, once you feel
00:34:03.290 --> 00:34:06.740
that you have a polished version
of the lecture, write up,
00:34:06.740 --> 00:34:09.810
ideally within four
days of the lecture--
00:34:09.810 --> 00:34:12.230
so that in terms of
expectations and timelines,
00:34:12.230 --> 00:34:14.480
again all of this
information is online--
00:34:14.480 --> 00:34:17.150
so you're finished with
polishing your lecture notes,
00:34:17.150 --> 00:34:20.120
within four days send me an
email, so both co-authors
00:34:20.120 --> 00:34:22.310
if there are two of
you, and I will schedule
00:34:22.310 --> 00:34:25.100
an appointment,
about half an hour,
00:34:25.100 --> 00:34:26.600
where I will sit
down with you to go
00:34:26.600 --> 00:34:29.630
through what you've written
and tell you some comments.
00:34:29.630 --> 00:34:31.790
So you can go back
and polish it further.
00:34:31.790 --> 00:34:34.699
And hopefully, that will
just be a one round thing.
00:34:34.699 --> 00:34:37.670
If more rounds are needed,
well, it's not ideal,
00:34:37.670 --> 00:34:42.199
but we'll make it happen until
the notes are ready to use
00:34:42.199 --> 00:34:44.429
for future generations.
00:34:44.429 --> 00:34:50.150
OK, any questions about any
of the course logistics?
00:34:50.150 --> 00:34:53.699
All right, so in the second
half of today's lecture,
00:34:53.699 --> 00:34:57.630
I want to take you through
a tour of modern additive
00:34:57.630 --> 00:34:58.830
combinatorics.
00:34:58.830 --> 00:35:01.320
And this is an area
of research which
00:35:01.320 --> 00:35:03.000
I am actively involved in.
00:35:03.000 --> 00:35:05.130
And it's something that
I am quite excited about.
00:35:05.130 --> 00:35:08.010
And part of the reason
why I teach this course--
00:35:08.010 --> 00:35:10.980
this course is something that
I developed a couple years ago
00:35:10.980 --> 00:35:12.810
when I taught for
the first time then--
00:35:12.810 --> 00:35:14.730
because I want to
introduce you guys
00:35:14.730 --> 00:35:20.370
to this very active and
exciting area of research.
00:35:20.370 --> 00:35:25.040
Now, what is added
combinatorics?
00:35:25.040 --> 00:35:27.360
The term itself is
actually fairly new.
00:35:27.360 --> 00:35:30.450
So the term, additive
combinatorics, I believe
00:35:30.450 --> 00:35:35.190
was coined by Terry Tao back
in the early 2000s as somewhat
00:35:35.190 --> 00:35:39.990
of a rebranding of an area that
already existed, but then got
00:35:39.990 --> 00:35:43.390
a lot of exciting developments
in the early 2000s.
00:35:43.390 --> 00:35:47.460
It's a deep and far reaching
subject with many connections
00:35:47.460 --> 00:35:51.600
to areas like graph theory,
harmonic analysis, or Fourier
00:35:51.600 --> 00:35:56.730
analysis, ergodic theory,
discrete geometry, logic
00:35:56.730 --> 00:36:00.540
and model theory, and has many
connections all over the place,
00:36:00.540 --> 00:36:03.000
and also has many deep theorems.
00:36:03.000 --> 00:36:05.880
So let me take you through a
tour historically of, I think,
00:36:05.880 --> 00:36:08.700
some of the major
milestones and landmarks
00:36:08.700 --> 00:36:12.702
in additive combinatorics.
00:36:12.702 --> 00:36:15.060
So after Schur's theorem,
which we discussed
00:36:15.060 --> 00:36:19.370
in the first half
of today's lecture,
00:36:19.370 --> 00:36:24.640
the next big result I would say
is Van der Waerden's theorem,
00:36:24.640 --> 00:36:30.210
which was 1927.
00:36:30.210 --> 00:36:36.390
Van der Waerden's theorem
says that every coloring
00:36:36.390 --> 00:36:49.610
of the positive integers
using finite many colors
00:36:49.610 --> 00:36:55.029
contains arbitrarily long
arithmetic progressions.
00:37:06.010 --> 00:37:09.050
So we'll see arithmetic
progressions come up a lot.
00:37:09.050 --> 00:37:12.730
So from now on we'll
abbreviate this word by AP.
00:37:12.730 --> 00:37:17.420
So AP stands for
Arithmetic Progressions.
00:37:17.420 --> 00:37:19.450
So instead of Schur's
theorem where you just
00:37:19.450 --> 00:37:22.070
find a single solution
to x plus y equals to z,
00:37:22.070 --> 00:37:26.020
so now, we're finding a
much bigger structure.
00:37:26.020 --> 00:37:28.750
Keep in mind, so a
novice mistake people
00:37:28.750 --> 00:37:32.440
make is to confuse arbitrarily
long arithmetic progressions
00:37:32.440 --> 00:37:33.580
with infinitely long.
00:37:33.580 --> 00:37:35.562
So these are definitely
not the same.
00:37:35.562 --> 00:37:36.520
So you can think about.
00:37:36.520 --> 00:37:39.310
I'll leave it to you as
an exercise, well, also
00:37:39.310 --> 00:37:43.270
homework exercise, that you can
color the integers with just
00:37:43.270 --> 00:37:45.820
two colors in a
way that destroys
00:37:45.820 --> 00:37:49.570
all possible infinitely long
monochromatic arithmetic
00:37:49.570 --> 00:37:51.130
progressions.
00:37:51.130 --> 00:37:53.980
So arbitrarily long is very
different from infinitely long.
00:37:53.980 --> 00:38:00.100
Now, so this was a great result,
but it provokes more questions.
00:38:00.100 --> 00:38:12.020
So Erdos-Turan in the
'30s, they asked--
00:38:12.020 --> 00:38:16.490
well, they conjectured
that the true reason in Van
00:38:16.490 --> 00:38:19.970
der Waerden's theorem of having
long arithmetic progressions,
00:38:19.970 --> 00:38:22.340
it's not so much
that you're coloring.
00:38:22.340 --> 00:38:25.550
It's just because if you
use finitely many colors,
00:38:25.550 --> 00:38:30.110
then one of the color classes
must have fairly high density.
00:38:30.110 --> 00:38:33.800
So one of the classes if you use
r colors has density at least 1
00:38:33.800 --> 00:38:34.940
over r.
00:38:34.940 --> 00:38:43.000
And they conjectured
that every subset
00:38:43.000 --> 00:38:54.400
of the positive integers, or the
integers with positive density,
00:38:54.400 --> 00:39:02.040
contains long--
00:39:02.040 --> 00:39:05.827
so arbitrarily long
arithmetic progressions.
00:39:08.373 --> 00:39:10.040
You may ask, what
does it mean, density?
00:39:10.040 --> 00:39:12.910
So you can define density
in many different ways.
00:39:12.910 --> 00:39:14.770
And it doesn't
actually really matter
00:39:14.770 --> 00:39:16.450
that much which
definition you use.
00:39:16.450 --> 00:39:19.280
But let me write
down one definition.
00:39:19.280 --> 00:39:24.100
So you can define given
a subset of integers
00:39:24.100 --> 00:39:27.100
the upper density, or
rather, let me just
00:39:27.100 --> 00:39:38.680
say that it has positive
upper density, if when we take
00:39:38.680 --> 00:39:44.450
the lim sup as n goes to
infinity and look at we'll
00:39:44.450 --> 00:39:48.980
take a scaling window and look
at what fraction of that window
00:39:48.980 --> 00:40:02.700
is a, then this number,
this limit sup is positive.
00:40:02.700 --> 00:40:05.340
So that's one definition
of positive density.
00:40:05.340 --> 00:40:08.075
There are many other
definitions, sometimes known
00:40:08.075 --> 00:40:09.470
as the Banach density.
00:40:09.470 --> 00:40:11.830
And you can take variations.
00:40:11.830 --> 00:40:13.680
I mean, for the purpose
of this discussion,
00:40:13.680 --> 00:40:15.350
they're all roughly equivalent.
00:40:15.350 --> 00:40:19.230
So let's not worry too
much about which definition
00:40:19.230 --> 00:40:22.150
of density we use here.
00:40:22.150 --> 00:40:24.550
All right, so Erdos
and Turan conjectured
00:40:24.550 --> 00:40:26.820
that the true reason for
Van der Waerden's theorem
00:40:26.820 --> 00:40:30.480
is that one of the color
classes has positive density.
00:40:30.480 --> 00:40:34.530
And this turned out to be an
amazingly prescient question
00:40:34.530 --> 00:40:39.120
and that one had to
wait several decades.
00:40:39.120 --> 00:40:43.710
So this conjecture was
made in the '30s, in 1936.
00:40:43.710 --> 00:40:46.350
So you had to wait several
decades before finding out
00:40:46.350 --> 00:40:48.100
what the answer is.
00:40:48.100 --> 00:40:52.870
So in a foundational
theorem, in the subject
00:40:52.870 --> 00:40:55.330
known as Roth's theorem--
00:40:55.330 --> 00:40:58.278
so Roth proved it in the '50s.
00:40:58.278 --> 00:40:58.820
I think '53--
00:41:13.730 --> 00:41:16.520
Roth proved that, I
think, '53, in the '50s--
00:41:16.520 --> 00:41:19.590
that k equals to 3 is true.
00:41:19.590 --> 00:41:25.330
So if I say that it
contains k term, arithmetic
00:41:25.330 --> 00:41:28.610
progressions for every k.
00:41:28.610 --> 00:41:31.180
And Roth proved that
every positive density
00:41:31.180 --> 00:41:33.340
subset contains a 3-term
arithmetic progression.
00:41:33.340 --> 00:41:37.160
And already, Roth introduced
very important ideas
00:41:37.160 --> 00:41:40.427
that we will see in this
course in two different forms.
00:41:40.427 --> 00:41:42.010
So in the first half
the course, we'll
00:41:42.010 --> 00:41:44.860
see a graph theoretic
proof that was found later
00:41:44.860 --> 00:41:47.433
in the '70s of Roth's theorem.
00:41:47.433 --> 00:41:48.850
And then in the
second half, we'll
00:41:48.850 --> 00:41:52.800
see Roth's original proof
that used Fourier analysis.
00:41:52.800 --> 00:41:55.220
So Fourier analysis
in number theory
00:41:55.220 --> 00:41:57.620
is also known as the
Hardy-Littlewood circle method.
00:41:57.620 --> 00:42:00.230
It's a powerful method in
analytic number theory.
00:42:00.230 --> 00:42:04.730
But there are very interesting
new ideas introduced by Roth as
00:42:04.730 --> 00:42:10.900
well in developing this result.
00:42:10.900 --> 00:42:14.615
The full conjecture was
settled by Szemeredi.
00:42:17.663 --> 00:42:19.080
It took another
couple of decades.
00:42:22.800 --> 00:42:29.980
So in the late '70s, Szemeredi
proved his landmark theorem
00:42:29.980 --> 00:42:34.330
that confirmed the
Erdos-Turan conjecture.
00:42:34.330 --> 00:42:36.600
Szemeredi's theorem
is a deep theorem.
00:42:36.600 --> 00:42:39.090
So this theorem
is the proof, what
00:42:39.090 --> 00:42:43.500
the original combinatorial
proof is a tour de force.
00:42:43.500 --> 00:42:46.290
And you can look
at the introduction
00:42:46.290 --> 00:42:51.600
of his paper, where there is
an enormously complex diagram--
00:42:51.600 --> 00:42:53.490
so you can see this
in the course notes--
00:42:53.490 --> 00:42:57.270
that lays out the logical
dependencies of all the lemmas
00:42:57.270 --> 00:42:59.040
and propositions in his paper.
00:42:59.040 --> 00:43:02.280
And even if you assume every
single statement is true,
00:43:02.280 --> 00:43:04.650
looking at that diagram,
it's not immediately clear
00:43:04.650 --> 00:43:07.170
what is going on because
the logical dependencies are
00:43:07.170 --> 00:43:08.740
so involved.
00:43:08.740 --> 00:43:14.320
So this was a really
complex proof.
00:43:14.320 --> 00:43:17.910
But not only that, Szemeredi's
theorem actually motivated
00:43:17.910 --> 00:43:20.030
a lot of subsequent research.
00:43:20.030 --> 00:43:23.700
So later on, researchers
from other areas
00:43:23.700 --> 00:43:27.720
came in and found also
sophisticated proofs
00:43:27.720 --> 00:43:32.230
of Szemeredi's theorem
from other areas
00:43:32.230 --> 00:43:34.470
and using other
tools, including--
00:43:34.470 --> 00:43:37.140
and here are some of the
most important perspectives,
00:43:37.140 --> 00:43:39.180
later perspectives, of
Szemeredi's theorem.
00:43:39.180 --> 00:43:44.630
So there was a proof
using ergodic theory
00:43:44.630 --> 00:43:48.120
that followed fairly
shortly after Szemeredi's
00:43:48.120 --> 00:43:50.810
original proof.
00:43:50.810 --> 00:43:52.170
This is due to Furstenberg.
00:43:54.780 --> 00:43:57.890
And initially, it wasn't clear,
because all of these proofs
00:43:57.890 --> 00:43:59.010
were so involved.
00:43:59.010 --> 00:44:03.890
It wasn't clear if the ergodic
theoretic proof was genuinely
00:44:03.890 --> 00:44:08.390
something new, or it was a
rephrasing of Szemeredi's
00:44:08.390 --> 00:44:10.250
combinatorial proof.
00:44:10.250 --> 00:44:12.050
But then very quickly
it was realized
00:44:12.050 --> 00:44:15.830
that there were extensions
of Szemeredi's theorem,
00:44:15.830 --> 00:44:19.280
other combinatorial results
that the ergodic theorists could
00:44:19.280 --> 00:44:23.450
establish using their methods,
so using the same methods
00:44:23.450 --> 00:44:25.220
or extensions of
the same methods
00:44:25.220 --> 00:44:29.440
that combinatorialists
did not know how to do.
00:44:29.440 --> 00:44:31.420
And to this date,
there are still
00:44:31.420 --> 00:44:34.690
theorems for which
the only known proofs
00:44:34.690 --> 00:44:37.812
use ergodic theory,
so extensions
00:44:37.812 --> 00:44:38.770
of Szemeredi's theorem.
00:44:38.770 --> 00:44:40.600
And I will mention
one later on today.
00:44:44.770 --> 00:44:47.020
So that's one of
the perspectives.
00:44:47.020 --> 00:44:50.710
The other perspective that
was also quite influential
00:44:50.710 --> 00:44:52.750
there is something known
as higher order Fourier
00:44:52.750 --> 00:45:07.000
analysis, which was pioneered
by Tim Gowers' in around 2000.
00:45:07.000 --> 00:45:10.840
So Gowers won the
Fields Medal, party
00:45:10.840 --> 00:45:12.850
for his work on
Banach spaces but also
00:45:12.850 --> 00:45:15.910
party for this development.
00:45:15.910 --> 00:45:18.520
So higher order Fourier
analysis is in some sense
00:45:18.520 --> 00:45:20.800
an extension of Roth's theorem.
00:45:20.800 --> 00:45:23.140
So anyway, Roth also
won a Fields Medal,
00:45:23.140 --> 00:45:25.330
although this is not
his most famous term.
00:45:25.330 --> 00:45:27.700
I'll say his second
most famous theorem.
00:45:27.700 --> 00:45:31.450
So Roth used this
Fourier analysis
00:45:31.450 --> 00:45:36.000
in the sense of Hardy-Littlewood
to control 3-term arithmetic
00:45:36.000 --> 00:45:37.140
progressions.
00:45:37.140 --> 00:45:39.240
But it turns out
that that method
00:45:39.240 --> 00:45:41.820
for very good
fundamental reasons
00:45:41.820 --> 00:45:45.930
completely fails for 4-term
arithmetic progressions.
00:45:45.930 --> 00:45:49.600
So we'll see later in the
course why that's the case,
00:45:49.600 --> 00:45:52.080
why is it that you cannot do
Fourier analysis to control
00:45:52.080 --> 00:45:53.640
4-term APs.
00:45:53.640 --> 00:45:56.970
But Gowers managed to find a
way to overcome that difficulty.
00:45:56.970 --> 00:45:58.800
And he came up
with an extension,
00:45:58.800 --> 00:46:00.810
with a generalization
of Fourier analysis,
00:46:00.810 --> 00:46:03.360
very powerful, very
difficult to use, actually.
00:46:03.360 --> 00:46:08.380
But that allows you to
understand longer arithmetic
00:46:08.380 --> 00:46:11.570
progressions.
00:46:11.570 --> 00:46:15.410
Another very
influential approach
00:46:15.410 --> 00:46:17.599
is called hypergraph regularity.
00:46:23.090 --> 00:46:26.130
So the hypergraph
regularity method
00:46:26.130 --> 00:46:36.190
was also discovered in the early
2000s independently by a team
00:46:36.190 --> 00:46:40.830
led by Rodl and also by Gowers.
00:46:44.170 --> 00:46:47.220
So the hypergraph
regularity method
00:46:47.220 --> 00:46:51.130
is an extension of what's known
as Szemeredi's regularity,
00:46:51.130 --> 00:46:53.740
Szemeredi's graph
regularity method.
00:46:53.740 --> 00:46:55.360
And this is the
method that will be
00:46:55.360 --> 00:47:00.100
a central topic in the
first half of this course.
00:47:00.100 --> 00:47:04.120
And it's a method that is
quite central, or at least some
00:47:04.120 --> 00:47:07.680
of the ideas quite central,
to Szemeredi's method.
00:47:07.680 --> 00:47:09.900
And he gave an
alternative proof.
00:47:09.900 --> 00:47:12.970
He and Ruzsa gave an alternative
proof of Roth's theorem
00:47:12.970 --> 00:47:15.030
using graph theory.
00:47:15.030 --> 00:47:17.210
And for a long time,
people realized
00:47:17.210 --> 00:47:20.630
that one could extend some of
those ideas to hypergraphs.
00:47:20.630 --> 00:47:23.570
But working out how
that proof goes actually
00:47:23.570 --> 00:47:25.730
took an enormous amount
of time and effort
00:47:25.730 --> 00:47:31.030
and resulted in this amazing
theorem on hypergraph.
00:47:31.030 --> 00:47:34.380
Let me mention these are
not the only methods that
00:47:34.380 --> 00:47:36.630
were used to extend
Szemeredi's theorem
00:47:36.630 --> 00:47:38.040
or give alternate proofs.
00:47:38.040 --> 00:47:39.790
There are many others.
00:47:39.790 --> 00:47:41.850
For example, you may
have heard of something
00:47:41.850 --> 00:47:44.110
called the polymath project.
00:47:44.110 --> 00:47:46.530
Raise your hand if you heard
of the polymath project.
00:47:46.530 --> 00:47:47.070
OK, great.
00:47:47.070 --> 00:47:49.920
So maybe about half of you.
00:47:49.920 --> 00:47:54.030
So this is an online
collaborative project
00:47:54.030 --> 00:48:00.150
started by Tim Gowers and also
famous people like Terry Tao.
00:48:00.150 --> 00:48:04.170
And they were all quite involved
in various polymath projects.
00:48:04.170 --> 00:48:07.800
And the first successful
polymath project
00:48:07.800 --> 00:48:11.670
produced a combinatorial
proof of something
00:48:11.670 --> 00:48:13.860
known as the density
Hales-Jewett theorem.
00:48:20.450 --> 00:48:22.840
So I won't explain
what it this here.
00:48:22.840 --> 00:48:26.120
So it's something which
is related to tic tac toe.
00:48:26.120 --> 00:48:27.770
But let me not go into that.
00:48:27.770 --> 00:48:32.260
So it's a deep combinatorial
theorem that had they
00:48:32.260 --> 00:48:34.480
known earlier using
ergodic theoretic methods,
00:48:34.480 --> 00:48:36.350
but they gave a new
combinatorial proof,
00:48:36.350 --> 00:48:41.200
in particular gave some
concrete bounds on this theorem
00:48:41.200 --> 00:48:45.760
and that in particular also
implies Szemeredi's theorem.
00:48:45.760 --> 00:48:47.500
So this gave a new proof.
00:48:47.500 --> 00:48:50.140
And as a result, they--
00:48:50.140 --> 00:48:52.020
it's an online
collaborative project--
00:48:52.020 --> 00:48:55.840
so they published this paper
under the pseudonym DHJ
00:48:55.840 --> 00:49:03.452
Polymath, where DHJ stands
for Density Hales Jewett.
00:49:03.452 --> 00:49:04.910
And they kept the
same name for all
00:49:04.910 --> 00:49:06.740
of the subsequent
papers published
00:49:06.740 --> 00:49:10.440
by the polymath project.
00:49:10.440 --> 00:49:14.180
So as you see through
all of these examples
00:49:14.180 --> 00:49:17.750
that there a lot of
work that were motivated
00:49:17.750 --> 00:49:19.370
by Szemeredi's theorem.
00:49:19.370 --> 00:49:21.460
This is truly a
foundational result,
00:49:21.460 --> 00:49:25.190
a foundational
theorem that gave way
00:49:25.190 --> 00:49:28.770
to a lot of important research.
00:49:28.770 --> 00:49:32.540
And Szemeredi himself
received an Apple Prize
00:49:32.540 --> 00:49:36.162
for his seminal contributions
to combinatorics and also
00:49:36.162 --> 00:49:37.370
theoretical computer science.
00:49:40.460 --> 00:49:42.980
We still don't
understand in some sense
00:49:42.980 --> 00:49:44.960
completely what
Szemeredi's theorem--
00:49:44.960 --> 00:49:48.410
you know, for example, we do
understand the optimal bounds.
00:49:48.410 --> 00:49:50.660
And also more importantly,
conceptually, we
00:49:50.660 --> 00:49:53.720
don't really understand
how these methods are
00:49:53.720 --> 00:49:55.740
related to each other.
00:49:55.740 --> 00:49:57.680
So there's some vague
sense that they all
00:49:57.680 --> 00:50:00.100
have some common things.
00:50:00.100 --> 00:50:06.340
But there is a lot of mystery as
to what do these methods coming
00:50:06.340 --> 00:50:08.460
from very different areas--
00:50:08.460 --> 00:50:10.278
ergodic theory,
harmonic analysis, you
00:50:10.278 --> 00:50:12.070
what do they all have
to do with each other
00:50:12.070 --> 00:50:14.260
but there is central theme.
00:50:14.260 --> 00:50:20.110
And this is also going to be
a theme in this course, which
00:50:20.110 --> 00:50:21.220
goes under the name--
00:50:21.220 --> 00:50:24.520
and I believe Terry Tao is the
one who popularized this name--
00:50:24.520 --> 00:50:35.420
the dichotomy between structure
and randomness, structure
00:50:35.420 --> 00:50:37.387
and pseudo randomness.
00:50:42.060 --> 00:50:46.930
Somehow it's a really fancy way
of saying signal versus noise.
00:50:46.930 --> 00:50:51.640
So I give you some object, I
give you some complex object,
00:50:51.640 --> 00:50:53.740
and there is some
mathematical way
00:50:53.740 --> 00:50:59.980
to separate the structure
from some noisy aspects, which
00:50:59.980 --> 00:51:01.690
behave random-like.
00:51:01.690 --> 00:51:03.760
So there will be many
places in this course
00:51:03.760 --> 00:51:09.210
where this dichotomy will
play an important role.
00:51:09.210 --> 00:51:11.010
Any questions at this point?
00:51:20.030 --> 00:51:24.020
I want to take you through some
generalizations and extensions
00:51:24.020 --> 00:51:26.630
of Szemeredi's theorem.
00:51:26.630 --> 00:51:28.940
So first, let's
look at what happens
00:51:28.940 --> 00:51:31.100
if we go to higher dimensions.
00:51:34.970 --> 00:51:39.100
Suppose we have a
subset in D dimensions,
00:51:39.100 --> 00:51:41.190
d-dimensional lattice.
00:51:41.190 --> 00:51:44.250
So we can also define
some notion of density.
00:51:44.250 --> 00:51:46.200
Again, it doesn't
matter precisely what
00:51:46.200 --> 00:51:47.680
is the notion you use.
00:51:47.680 --> 00:51:57.520
For example, we can say that
it has a positive upper density
00:51:57.520 --> 00:52:12.882
if this lim sup is positive.
00:52:16.260 --> 00:52:19.460
So Szemeredi's theorem
in one dimension
00:52:19.460 --> 00:52:22.335
tells us that if you have some
sort of positive density, then
00:52:22.335 --> 00:52:26.550
I can find arbitrarily long
arithmetic progressions.
00:52:26.550 --> 00:52:28.860
So what should the
corresponding generalization
00:52:28.860 --> 00:52:31.760
in higher dimensions?
00:52:31.760 --> 00:52:34.350
Well, here's a notion
that I can define, namely
00:52:34.350 --> 00:52:55.030
that we say that a contains
arbitrary constellations
00:52:55.030 --> 00:52:57.972
to mean that--
00:52:57.972 --> 00:52:58.930
so what does that mean?
00:52:58.930 --> 00:53:00.555
So a constellation,
you can think of it
00:53:00.555 --> 00:53:04.920
as some finite pattern, so a
set of stars in the sky, so
00:53:04.920 --> 00:53:05.500
some pattern.
00:53:05.500 --> 00:53:09.970
And I want to find that
pattern somewhere in a, where
00:53:09.970 --> 00:53:11.360
I'm allowed to dilate.
00:53:11.360 --> 00:53:16.970
So I'm allowed to do to
multiply pattern by some number
00:53:16.970 --> 00:53:18.206
and also translate.
00:53:18.206 --> 00:53:20.320
So on the finite pattern--
00:53:20.320 --> 00:53:27.150
so what I mean precisely is
that for every finite subset
00:53:27.150 --> 00:53:40.420
of the grid, there exists some
translation and some dilation,
00:53:40.420 --> 00:53:47.200
such that once I apply this
dilation and translation
00:53:47.200 --> 00:53:56.490
to my pattern F, meaning I'm
looking at the image of this F
00:53:56.490 --> 00:54:00.730
under this transformation,
then this set lies inside a.
00:54:03.901 --> 00:54:07.030
So you see that
arithmetic progressions
00:54:07.030 --> 00:54:11.040
is the constellation,
just numbers 1 through k.
00:54:14.020 --> 00:54:15.822
So that's a definition.
00:54:15.822 --> 00:54:17.780
And the multi-dimensional
Szemeredi's theorem--
00:54:26.040 --> 00:54:27.910
so the multi-dimensional
generalization
00:54:27.910 --> 00:54:37.500
of Szemeredi's theorem says
that for every subset--
00:54:37.500 --> 00:54:43.980
so every subset of the
d-dimensional lattice
00:54:43.980 --> 00:54:54.350
of possible density contains
arbitrary constellations.
00:55:04.170 --> 00:55:06.540
You give me a pattern,
and I can find
00:55:06.540 --> 00:55:11.180
this pattern inside a, provided
that a has positive density.
00:55:14.350 --> 00:55:21.010
So in particular, if I want to
find a 10 by 10 square grid,
00:55:21.010 --> 00:55:26.910
so meaning suppose I want to
find a pattern which consists
00:55:26.910 --> 00:55:35.380
of something like that, a
10 by 10 square grid, where
00:55:35.380 --> 00:55:38.710
all of these lengths
are equal, but I
00:55:38.710 --> 00:55:40.000
don't specify what they are.
00:55:40.000 --> 00:55:43.450
But as long as they are equal,
then the theorem tells me
00:55:43.450 --> 00:55:46.810
that as long as a
has positive density,
00:55:46.810 --> 00:55:51.100
then I can find such
a pattern inside a.
00:55:51.100 --> 00:55:55.300
So this theorem was proved
by Furstenberg and Ketsen.
00:55:59.860 --> 00:56:03.530
So you see that it is a
generalization of Szemeredi's.
00:56:03.530 --> 00:56:06.530
So the one-dimensional case is
precisely Szemeredi's theorem.
00:56:10.490 --> 00:56:13.580
So Furstenberg and Ketsen,
using ergodic theory
00:56:13.580 --> 00:56:18.510
showed that one can
generalize Szemeredi's theorem
00:56:18.510 --> 00:56:20.740
to the multi-dimensional
setting.
00:56:20.740 --> 00:56:23.610
However, the
combinatorial approaches
00:56:23.610 --> 00:56:26.680
employed by Szemeredi did
not easily generalize.
00:56:26.680 --> 00:56:30.600
So it took another couple
of decades at least
00:56:30.600 --> 00:56:33.840
for people to find a
combinatorial proof
00:56:33.840 --> 00:56:36.780
of this result. And
namely that happened
00:56:36.780 --> 00:56:40.750
with the hypergraph
regularity method.
00:56:40.750 --> 00:56:44.305
So this was one of the
motivations of this project.
00:56:44.305 --> 00:56:45.680
And you say, OK,
what's the point
00:56:45.680 --> 00:56:46.805
of having different proofs?
00:56:46.805 --> 00:56:50.540
Well, for one thing it's nice
to know different perspectives
00:56:50.540 --> 00:56:52.820
to important theorem.
00:56:52.820 --> 00:56:56.610
But there's also
concrete objective.
00:56:56.610 --> 00:57:00.150
In particular, it turns out
that if you prove something
00:57:00.150 --> 00:57:04.100
using ergodic theory, because--
00:57:04.100 --> 00:57:06.690
we will not discuss ergodic
theory in this course.
00:57:06.690 --> 00:57:10.550
But roughly, one of the
early steps in such a proof
00:57:10.550 --> 00:57:13.330
applies compactness.
00:57:13.330 --> 00:57:16.150
And that already
destroys any chance
00:57:16.150 --> 00:57:18.940
of getting concrete
quantitative bounds.
00:57:18.940 --> 00:57:23.920
So you can ask if I want
to find a 10 by 10 pattern
00:57:23.920 --> 00:57:32.090
and I have density 1%, how
large do I need to look?
00:57:32.090 --> 00:57:34.980
How far do I have to look in
order to find that pattern?
00:57:34.980 --> 00:57:36.710
So that's a
quantitative question
00:57:36.710 --> 00:57:39.790
that is actually not at all
addressed by ergodic theory.
00:57:39.790 --> 00:57:44.500
So the later methods using
combinatorial methods
00:57:44.500 --> 00:57:46.410
gave you concrete bounds.
00:57:46.410 --> 00:57:49.310
And so there are some
concrete differences
00:57:49.310 --> 00:57:50.380
between these methods.
00:57:53.930 --> 00:57:57.040
So this theorem
reminds me of the scene
00:57:57.040 --> 00:57:59.400
from the movie a
Beautiful Mind, which
00:57:59.400 --> 00:58:02.700
is one of the greatest
mathematical movies
00:58:02.700 --> 00:58:04.450
in some sense.
00:58:04.450 --> 00:58:08.020
And so there's a scene
there where Russell Crowe
00:58:08.020 --> 00:58:11.640
playing John Nash--
00:58:11.640 --> 00:58:13.730
so there were at
this fancy party.
00:58:13.730 --> 00:58:17.780
And Nash was with his
soon to be wife, Alicia.
00:58:17.780 --> 00:58:21.670
And he points to the sky
and tells her, pick a shape.
00:58:21.670 --> 00:58:24.683
Pick a shape and I can find
for you among the stars.
00:58:24.683 --> 00:58:26.850
And so this is what the
theorem allows you to do it.
00:58:31.340 --> 00:58:33.540
So give me a shape
and I can find
00:58:33.540 --> 00:58:35.230
that constellation inside a.
00:58:38.720 --> 00:58:41.464
Let's look at other
generalizations.
00:58:44.430 --> 00:58:47.120
So far, we are looking
at linear patterns.
00:58:47.120 --> 00:58:50.070
So we're looking at linear
dilations and translations.
00:58:50.070 --> 00:58:53.040
But what about
polynomial patterns?
00:58:53.040 --> 00:58:54.450
So here's a question.
00:58:54.450 --> 00:58:58.230
Suppose I give you a dense
subset, a positive density
00:58:58.230 --> 00:59:00.240
subset of integers.
00:59:00.240 --> 00:59:06.590
Can you find two numbers whose
difference is a perfect square?
00:59:06.590 --> 00:59:08.470
So this question
was asked by Lovasz.
00:59:08.470 --> 00:59:14.310
And a positive answer was given
in the late '70s by Furstenberg
00:59:14.310 --> 00:59:16.749
and Sarkozy independently.
00:59:28.485 --> 00:59:33.460
So Furstenberg and Sarkozy,
they showed using different
00:59:33.460 --> 00:59:37.280
methods-- so one ergodic
theoretic and the other is more
00:59:37.280 --> 00:59:39.340
harmonic analytic--
00:59:39.340 --> 00:59:47.110
that every subset of the
integers, so every subset
00:59:47.110 --> 00:59:56.460
of positive integers,
with positive density
00:59:56.460 --> 01:00:04.530
contains two numbers
differing by a perfect square.
01:00:14.120 --> 01:00:15.800
So in other words,
we can always find
01:00:15.800 --> 01:00:19.340
the pattern x plus y squared.
01:00:22.810 --> 01:00:24.670
So what about other
polynomial patterns?
01:00:24.670 --> 01:00:27.130
Instead of this y
squared, suppose you just
01:00:27.130 --> 01:00:30.820
give me some other
polynomial or maybe
01:00:30.820 --> 01:00:33.500
a collection of polynomials.
01:00:33.500 --> 01:00:34.960
So what can I say?
01:00:34.960 --> 01:00:38.110
Well, there are some things
for which this is not true.
01:00:38.110 --> 01:00:40.210
Can you give me an
example where if I
01:00:40.210 --> 01:00:42.852
putting the wrong
polynomial it's not true?
01:00:46.120 --> 01:00:48.100
What if the polynomial
is the constant 1?
01:00:52.020 --> 01:00:55.880
If you take the even
numbers, has density 1/2,
01:00:55.880 --> 01:00:59.520
but it doesn't contain any
patterns of x and x plus 1.
01:00:59.520 --> 01:01:03.610
So I need to say some hypotheses
about these polynomials.
01:01:03.610 --> 01:01:07.200
So a vast generalization
of this result,
01:01:07.200 --> 01:01:17.420
so known as polynomial
Szemeredi theorem,
01:01:17.420 --> 01:01:30.340
says that if A is a subset of
integers with positive density,
01:01:30.340 --> 01:01:38.860
and if we have these
polynomials, P1 through Pk
01:01:38.860 --> 01:01:51.460
with integer coefficients
and zero constant terms,
01:01:51.460 --> 01:01:54.520
then I can always
find a pattern.
01:01:54.520 --> 01:02:02.200
So there exists some x
and positive integer y
01:02:02.200 --> 01:02:10.140
such that this pattern, x
plus P1 of y, x plus P2 of y,
01:02:10.140 --> 01:02:18.530
and so on x, plus Pk of
y, they all lie in A.
01:02:18.530 --> 01:02:22.630
So in other words, succinctly,
every subset of integers
01:02:22.630 --> 01:02:25.885
with positive density contains
arbitrary polynomial patterns.
01:02:29.240 --> 01:02:30.628
So this was proved--
01:02:30.628 --> 01:02:32.170
so this was an
important result proof
01:02:32.170 --> 01:02:41.090
by Bergelson and Liebman
using ergodic theory.
01:02:41.090 --> 01:02:44.340
And so far for this
general statement,
01:02:44.340 --> 01:02:49.382
the only known proof
uses ergodic theory.
01:02:49.382 --> 01:02:50.965
So there was some
recent developments,
01:02:50.965 --> 01:02:52.423
recent pretty
exciting developments
01:02:52.423 --> 01:02:55.280
that for some
specific cases where
01:02:55.280 --> 01:02:58.460
if you have some additional
restrictions on the P's, then
01:02:58.460 --> 01:03:01.130
there are other methods
coming from Fourier analytic,
01:03:01.130 --> 01:03:04.372
harmonic analytic methods
that could give you
01:03:04.372 --> 01:03:06.580
a different proof that allows
you to get some bounds.
01:03:06.580 --> 01:03:11.260
Remember, the ergodic
proof gives you no bounds.
01:03:11.260 --> 01:03:14.150
But so far, in general,
the only method known
01:03:14.150 --> 01:03:16.000
is ergodic theoretic.
01:03:16.000 --> 01:03:18.110
And actually,
Bergelson and Liebman
01:03:18.110 --> 01:03:21.080
proved something which is more
general than what I've stated.
01:03:21.080 --> 01:03:26.278
So this is also true in a
multidimensional setting.
01:03:26.278 --> 01:03:28.820
I won't state that precisely,
but you can imagine what it is.
01:03:33.270 --> 01:03:39.347
Let me mention one more theorem
that many of you I imagine
01:03:39.347 --> 01:03:39.930
have heard of.
01:03:42.343 --> 01:03:43.760
And this is the
Green Tao theorem.
01:03:48.853 --> 01:03:54.390
So the Green Tao theorem
says that the primes
01:03:54.390 --> 01:03:57.525
contain arbitrarily long
arithmetic progressions.
01:04:07.440 --> 01:04:09.210
So this is a famous theorem.
01:04:09.210 --> 01:04:12.840
And it's one of the
most celebrated results
01:04:12.840 --> 01:04:14.460
of the past couple of decades.
01:04:14.460 --> 01:04:16.660
And it resolved some
longstanding folklore
01:04:16.660 --> 01:04:19.350
conjectures in number theory.
01:04:19.350 --> 01:04:21.790
The Green Tao
theorem, well, you see
01:04:21.790 --> 01:04:25.480
that in form it looks somewhat
like Szemeredi 's theorem.
01:04:25.480 --> 01:04:28.110
But it doesn't follow
from Szemeredi 's theorem.
01:04:28.110 --> 01:04:30.555
Well, the primes, they
don't have positive density.
01:04:30.555 --> 01:04:31.930
The prime number
theorem tells us
01:04:31.930 --> 01:04:35.870
that density decays
like 1 over log n.
01:04:35.870 --> 01:04:39.200
So what about quantity versions
of Szemeredi 's theorem?
01:04:39.200 --> 01:04:40.910
It is possible.
01:04:40.910 --> 01:04:43.580
Although we do not know how
to prove such statement,
01:04:43.580 --> 01:04:46.370
it is possible that
a density of primes
01:04:46.370 --> 01:04:51.930
alone might guarantee the
Green Tao theorem in that it
01:04:51.930 --> 01:04:54.030
is possible that
Szemeredi 's theorem
01:04:54.030 --> 01:04:59.010
is true for any set
whose density decays
01:04:59.010 --> 01:05:02.100
like the prime numbers,
like 1 over log n.
01:05:02.100 --> 01:05:04.890
But no we're quite far from
proving such a statement.
01:05:04.890 --> 01:05:08.330
And that's not what
Green and Tao did.
01:05:08.330 --> 01:05:14.560
Instead, they took Szemeredi
's theorem as a black box
01:05:14.560 --> 01:05:19.520
and applied it to some
variant of the primes
01:05:19.520 --> 01:05:23.240
and showed that
inside this variant,
01:05:23.240 --> 01:05:25.070
Szemeredi 's theorem
is also true,
01:05:25.070 --> 01:05:28.890
and that the primes sit inside
this variant of the primes,
01:05:28.890 --> 01:05:32.880
known as pseudo primes, as a set
of relatively positive density,
01:05:32.880 --> 01:05:35.060
but somehow
transferring Szemeredi
01:05:35.060 --> 01:05:38.940
's theorem from the dense
setting to a sparser setting.
01:05:38.940 --> 01:05:42.110
So this is a very
exciting technique.
01:05:42.110 --> 01:05:46.160
And as a result, Green-Tao
proved not just that the primes
01:05:46.160 --> 01:05:49.340
contain arbitrarily long
arithmetic progressions,
01:05:49.340 --> 01:05:56.480
but every relatively dense,
so relatively positive
01:05:56.480 --> 01:06:05.120
density subset, of the
primes contains arbitrarily
01:06:05.120 --> 01:06:07.053
long arithmetic progressions.
01:06:11.170 --> 01:06:13.240
To prove this theorem
they incorporated
01:06:13.240 --> 01:06:17.850
many different ideas coming
from many different areas
01:06:17.850 --> 01:06:20.790
of mathematics, including
harmonic analysis,
01:06:20.790 --> 01:06:24.870
some ideas coming from
combinatorics, and number
01:06:24.870 --> 01:06:25.690
theory as well.
01:06:25.690 --> 01:06:28.710
So there were some innovations
at the time in number theory
01:06:28.710 --> 01:06:31.180
that were employed
in this result.
01:06:31.180 --> 01:06:34.350
So this is certainly
a landmark theorem.
01:06:34.350 --> 01:06:38.520
And although we will
not discuss a full proof
01:06:38.520 --> 01:06:41.680
of the Green-Tao theorem, we
will go into some of the ideas
01:06:41.680 --> 01:06:42.780
through this course.
01:06:42.780 --> 01:06:44.725
And I will show
you bits and pieces
01:06:44.725 --> 01:06:46.350
that we will see
throughout the course.
01:06:48.980 --> 01:06:53.740
So this is meant to be
a very fast tour of what
01:06:53.740 --> 01:06:57.550
happened in the last 100 years
in additive combinatorics,
01:06:57.550 --> 01:07:00.280
taking you from Schur's
theorem, which was really
01:07:00.280 --> 01:07:02.680
about 100 years
ago, to something
01:07:02.680 --> 01:07:05.710
that is much more modern.
01:07:05.710 --> 01:07:08.680
But now, instead of
being up in the stars,
01:07:08.680 --> 01:07:11.250
let's come back down to Earth.
01:07:11.250 --> 01:07:13.940
And I want to talk about
what we'll do next.
01:07:13.940 --> 01:07:16.720
So what are some of the
things that we can actually
01:07:16.720 --> 01:07:23.200
prove that doesn't
involve taking up 50 pages
01:07:23.200 --> 01:07:27.070
using a complex logical diagram,
as Szemeredi did in his paper.
01:07:27.070 --> 01:07:29.700
So what are some of the simple
things that we can start with?
01:07:29.700 --> 01:07:33.440
Well, so first, let's go
back to Roth's theorem.
01:07:33.440 --> 01:07:36.670
So Roth's theorem, we
stated it up there.
01:07:36.670 --> 01:07:39.385
But let me restate it
in a finitary form.
01:07:43.256 --> 01:07:45.580
So Roth's theorem
is the statement
01:07:45.580 --> 01:07:52.960
that every subset of
integers 1 through n that
01:07:52.960 --> 01:07:58.630
avoids 3-term
arithmetic progressions
01:07:58.630 --> 01:08:05.380
must have size little
o of N. So earlier we
01:08:05.380 --> 01:08:08.110
gave an infinitary
statement that if you
01:08:08.110 --> 01:08:10.470
have a positive density
subset of the integers
01:08:10.470 --> 01:08:12.340
that it contains
a three AP, this
01:08:12.340 --> 01:08:16.359
is an equivalent
finitary statement.
01:08:16.359 --> 01:08:19.930
Roth's original proof
used Fourier analysis.
01:08:19.930 --> 01:08:24.220
And a different proof was
given in the '70s by Rusza
01:08:24.220 --> 01:08:27.460
and Szemeredi using
graph theoretic methods.
01:08:27.460 --> 01:08:31.270
So how does graph theory
have to do with this result?
01:08:31.270 --> 01:08:34.060
And this shouldn't be
surprising to this point, given
01:08:34.060 --> 01:08:38.830
that we already saw how we
used Ramsay's theorem, graph
01:08:38.830 --> 01:08:40.750
theoretic result, to
prove Schur's theorem,
01:08:40.750 --> 01:08:42.800
which is something that
is number theoretic.
01:08:42.800 --> 01:08:46.029
So something similar happens.
01:08:46.029 --> 01:08:50.120
But now, the question is what
is the graph theoretic problem
01:08:50.120 --> 01:08:52.029
that we need to look at?
01:08:52.029 --> 01:08:56.000
So for Schur's theorem it was
Ramsey's theorem for triangles.
01:08:56.000 --> 01:08:58.750
But what about for
Roth's theorem?
01:08:58.750 --> 01:09:02.930
A naive guess is the following.
01:09:02.930 --> 01:09:05.050
So what's the question
that we should ask?
01:09:07.700 --> 01:09:11.930
Here's a somewhat naive
guess, which turns out
01:09:11.930 --> 01:09:13.670
not to be the right
question, but still
01:09:13.670 --> 01:09:15.649
an interesting
question, which is
01:09:15.649 --> 01:09:30.662
what is the maximum number of
edges in a triangle-free graph
01:09:30.662 --> 01:09:31.659
on n vertices?
01:09:37.149 --> 01:09:39.109
Now, this is not
totally a stupid guess,
01:09:39.109 --> 01:09:44.569
because as you imagine from what
we said with Schur's theorem,
01:09:44.569 --> 01:09:47.600
somehow you want to
set up a graph so
01:09:47.600 --> 01:09:50.760
that the triangles correspond
to the 3-term arithmetic
01:09:50.760 --> 01:09:52.100
progressions.
01:09:52.100 --> 01:09:53.930
And you want to set
it up in such a way
01:09:53.930 --> 01:09:57.890
that this question about what's
the maximum size subset of 1
01:09:57.890 --> 01:10:00.350
through n without
3 APs translates
01:10:00.350 --> 01:10:03.350
into some question about what's
the maximum number of edges
01:10:03.350 --> 01:10:07.155
in a graph that
has some property?
01:10:07.155 --> 01:10:09.220
So what is that property?
01:10:09.220 --> 01:10:12.120
So this is not a
totally stupid guess.
01:10:12.120 --> 01:10:16.320
But it turns out this
question is relatively easy.
01:10:16.320 --> 01:10:17.540
Still it has a name.
01:10:17.540 --> 01:10:23.570
So this was found by
Mantel about 100 years ago,
01:10:23.570 --> 01:10:24.790
so known as Mantel's theorem.
01:10:28.350 --> 01:10:32.432
And the answer, well,
we'll see a proof.
01:10:32.432 --> 01:10:34.390
So the first thing we'll
do in the next lecture
01:10:34.390 --> 01:10:39.210
is prove Mantel's theorem, but
I do want to hold suspense.
01:10:39.210 --> 01:10:40.900
I mean the answer,
it turns out to be
01:10:40.900 --> 01:10:42.490
fairly simple to describe.
01:10:42.490 --> 01:10:45.130
Namely that you split
the vertices into two
01:10:45.130 --> 01:10:46.930
basically equal halves.
01:10:46.930 --> 01:10:52.900
And you join all the possible
edges between the two halves.
01:10:52.900 --> 01:10:58.290
So this complete bipartide graph
with two equal-sized parts.
01:10:58.290 --> 01:11:01.438
And it turns out this graph,
you see this triangle-free
01:11:01.438 --> 01:11:03.730
and also turns out to have
the maximum number of edges.
01:11:03.730 --> 01:11:04.945
Yeah, question.
01:11:04.945 --> 01:11:07.735
AUDIENCE: What are asymptotics
for three arithmetic
01:11:07.735 --> 01:11:08.458
progression of--
01:11:08.458 --> 01:11:10.250
YUFEI ZHAO: Let me get
to that in a second.
01:11:10.250 --> 01:11:14.186
OK, so I'll talk about
asymptotics in a second.
01:11:14.186 --> 01:11:16.910
So it turns that this is not the
right graph theoretic question
01:11:16.910 --> 01:11:17.410
to ask.
01:11:17.410 --> 01:11:21.298
So what is the right graph
theoretic question to ask?
01:11:21.298 --> 01:11:22.340
I'll tell you what it is.
01:11:22.340 --> 01:11:25.210
I mean it shouldn't be
clear to you at this point.
01:11:25.210 --> 01:11:28.450
It still seems like an
interesting question,
01:11:28.450 --> 01:11:32.720
but it's also somewhat bizarre
to think about if you've never
01:11:32.720 --> 01:11:34.170
seen this before.
01:11:34.170 --> 01:11:39.750
So what is the maximum
number of edges
01:11:39.750 --> 01:11:53.290
in an n vertex graph, where
every edge lies in exactly one
01:11:53.290 --> 01:11:53.790
triangle?
01:12:03.270 --> 01:12:06.050
So I want a graph with
lots and lots of edges
01:12:06.050 --> 01:12:10.140
where every edge sits
in exactly one triangle.
01:12:10.140 --> 01:12:12.530
Now, you might have
some difficulty even
01:12:12.530 --> 01:12:15.110
coming up with good graphs
that have this property.
01:12:15.110 --> 01:12:15.890
And that's OK.
01:12:15.890 --> 01:12:19.290
These are very strange
things to think about.
01:12:19.290 --> 01:12:21.820
But we'll see many
examples of it later on.
01:12:21.820 --> 01:12:27.120
We'll also see
how Roth's theorem
01:12:27.120 --> 01:12:30.730
is connected to this
graph theoretic question.
01:12:30.730 --> 01:12:32.500
Just to give you
a hint, you know,
01:12:32.500 --> 01:12:35.050
where does exactly one
triangle come from,
01:12:35.050 --> 01:12:39.370
it's because even if you avoid
3-term arithmetic progressions,
01:12:39.370 --> 01:12:41.770
there are still these
trivial 3-term arithmetic
01:12:41.770 --> 01:12:45.910
progressions, where you keep
the same number three times.
01:12:45.910 --> 01:12:47.330
And in graph
theoretic world, that
01:12:47.330 --> 01:12:51.961
comes to the unique triangle
that every edge sits on.
01:12:55.120 --> 01:13:01.450
So to address the question
about quantitative bounds,
01:13:01.450 --> 01:13:06.080
for Roth's theorem, it
turns out that we have
01:13:06.080 --> 01:13:07.880
upper bounds and lower bounds.
01:13:07.880 --> 01:13:10.160
And it is still a
wide open question
01:13:10.160 --> 01:13:13.310
as to what these
things should be.
01:13:13.310 --> 01:13:16.820
And roughly speaking,
the best lower bound
01:13:16.820 --> 01:13:18.860
comes from a construction,
which we'll see later
01:13:18.860 --> 01:13:25.810
in this course, the higher size
around n divided by e to the c
01:13:25.810 --> 01:13:28.210
root log n.
01:13:28.210 --> 01:13:40.745
And the best upper bound is of
the form roughly n over log n.
01:13:40.745 --> 01:13:42.370
That's maybe a little
bit hard to think
01:13:42.370 --> 01:13:44.480
about how these numbers behave.
01:13:44.480 --> 01:13:47.590
So if you raise both
sides to-- the denominator
01:13:47.590 --> 01:13:50.970
to e to the something, then
it's maybe easier to compare.
01:13:50.970 --> 01:13:52.320
But it's still a pretty far gap.
01:13:52.320 --> 01:13:53.470
So still a pretty big gap.
01:13:56.360 --> 01:13:58.590
There's a famous conjecture
of Erdos some of you
01:13:58.590 --> 01:14:00.430
might have heard
of, that if you have
01:14:00.430 --> 01:14:02.290
a subset of the
positive integers
01:14:02.290 --> 01:14:05.440
with divergent
harmonic series, then
01:14:05.440 --> 01:14:10.090
it contains arbitrarily
long automatic progressions.
01:14:10.090 --> 01:14:11.830
That's a very
attractive statement.
01:14:11.830 --> 01:14:14.440
But somehow I don't like
the statement so much,
01:14:14.440 --> 01:14:17.140
because it seems to
make a too pretty.
01:14:17.140 --> 01:14:18.940
And the statement
really is about what
01:14:18.940 --> 01:14:24.430
is the bounds on Roth's theorem
and on Szemeredi 's theorem.
01:14:24.430 --> 01:14:27.610
And having divergent
harmonic series
01:14:27.610 --> 01:14:34.030
is roughly the same as trying
to prove Roth's theorem slightly
01:14:34.030 --> 01:14:36.490
better than the bound
that we currently have,
01:14:36.490 --> 01:14:38.500
somehow breaking this
logarithmic barrier.
01:14:38.500 --> 01:14:41.950
So that conjecture, that having
divergent harmonic series,
01:14:41.950 --> 01:14:45.548
implies 3-term
APs is still open.
01:14:45.548 --> 01:14:46.340
That is still open.
01:14:46.340 --> 01:14:48.940
So where the bound's very
close to what we can prove,
01:14:48.940 --> 01:14:51.160
but it is still open.
01:14:51.160 --> 01:14:54.710
For this question, we will
see later in this course,
01:14:54.710 --> 01:14:59.380
once we've developed Szemeredi
's regularity lemma that we
01:14:59.380 --> 01:15:07.310
can prove an upper bound of o
to the n squared, so little n.
01:15:07.310 --> 01:15:11.916
And that will suffice for
proving Roth's theorem.
01:15:11.916 --> 01:15:13.450
It turns out that
we don't know what
01:15:13.450 --> 01:15:14.680
the right answer should be.
01:15:14.680 --> 01:15:18.180
We don't know what is
the best such graph.
01:15:18.180 --> 01:15:20.790
And it turns out the
best construction
01:15:20.790 --> 01:15:25.590
for this graph there comes from
over here, the best lower bound
01:15:25.590 --> 01:15:28.710
construction of a
set, of a large set
01:15:28.710 --> 01:15:31.090
without 3-term
arithmetic progressions.
01:15:31.090 --> 01:15:34.350
So I'm giving you
a preview of more
01:15:34.350 --> 01:15:37.050
of these connections between
additive combinatorics on one
01:15:37.050 --> 01:15:40.370
hand and graph theory
on the other hand
01:15:40.370 --> 01:15:42.750
that we'll see
throughout this course.
01:15:45.690 --> 01:15:48.020
Any questions?
01:15:48.020 --> 01:15:48.560
OK.
01:15:48.560 --> 01:15:50.610
So just to tell you what's
going to happen next,
01:15:50.610 --> 01:15:52.820
so the next thing that
we're going to discuss
01:15:52.820 --> 01:15:55.480
is basically extremal
graph theory.
01:15:55.480 --> 01:15:59.360
And in particular, if you
forbid some structure,
01:15:59.360 --> 01:16:03.900
such as a triangle, maybe a four
cycle, maybe some other graph,
01:16:03.900 --> 01:16:07.020
what can you say about the
maximum number of edges?
01:16:07.020 --> 01:16:11.207
And there are still a lot of
interesting open problems,
01:16:11.207 --> 01:16:11.790
even for that.
01:16:11.790 --> 01:16:15.940
I forbid some H. What's the
maximum number of edges?
01:16:15.940 --> 01:16:20.360
So the next few lectures
will be on that topic.