WEBVTT
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YUFEI ZHAO: Last time, we
considered the relationship
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between pseudo-random graphs
and their eigenvalues.
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And the main message is that
the smaller your second largest
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eigenvalue is, the more
pseudo-random a graph is.
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In particular, we were
looking at this class
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of graphs that are d-regular--
they are somewhat easier
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to think about.
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And there is a
limit to how small
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the second largest
eigenvalue can be.
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And that was given by
the Alon-Boppana bound.
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You should think of d
here as a fixed number--
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so here, d is a fixed constant.
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Then, as the number of
vertices becomes large,
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the second largest eigenvalue
of a d-regular graph cannot be
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less than this
quantity over here.
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So this is the
limit to how small
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this second largest
eigenvalue can be.
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And last time, we gave
a proof of this bound
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by constructing an
appropriate function that
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witnesses this lambda 2.
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We also gave a
second proof which
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proves a slightly
weaker result, which
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is that the second largest
eigenvalue in absolute value
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is at least this quantity.
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So in spirit, it amounts to
roughly the same result--
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although technically,
it's a little bit weaker.
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And that one we proved
by counting walks.
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And also at the
end of last time,
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I remarked that
this number here--
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the fundamental
significance of this number
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is that it is the
spectral radius
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of the infinite d-regular tree.
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So that's why this
number is here.
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Of course, we proved
some lower bound.
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But you can always
ask the question,
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is this the best
possible lower bound?
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Maybe it's possible to prove
a somewhat higher bound.
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And that turns out
not to be the case.
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So that's the first
thing that we'll
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see today is some discussions--
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I won't show you any proofs--
but some discussions on why
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this number is best possible.
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And this is a very interesting
area of graph theory--
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goes under the name
of Ramanujan graphs.
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So I'll explain the history
in a second, why they're
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called Ramanujan graphs.
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Ramanujan did not
study these graphs,
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but they are called
them for good reasons.
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So by definition, a Ramanujan
graph is a d-regular graph,
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such that, if you look at its
eigenvalue of the adjacency
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matrix, as above, the second
largest eigenvalue in absolute
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value is, at most,
that bound up there--
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2 root d minus 1.
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So it's the best possible
constant you could put here
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so that there still exists
infinitely many d-regular
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Ramanujan graphs for fixed d--
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and the size of the
graph going to infinity.
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And the last time, we also
introduced some terminology.
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Let me just repeat that here.
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So this is, in other
words, an nd lambda graph,
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with lambda at most
2 root d minus 1.
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Now, it is not hard to obtain
a single example of a Ramanujan
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graph.
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So I just want some
graph such that--
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or the top eigenvalue is d.
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I want the other
ones to be small.
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So for example, if you get
this click, it's d-regular.
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Here, the top eigenvalue is d.
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And if it's not
too hard to compute
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that all the other eigenvalues
are equal to exactly minus 1.
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So this is an easy computation.
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But the point is that I
want to construct graphs--
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I want to understand
whether they
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are graphs where d is fixed.
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So this is somehow
not a good example.
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What we really want is fixed
d and n going to infinity.
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Large number of vertices.
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And the main open conjecture in
this area is that for every d,
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there exists infinitely many
Ramanujan d-regular graphs.
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So let me tell you
some partial results
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and also explain the
history of why they're
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called Ramanujan graphs.
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So the first paper
where this name
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appeared and coined this name--
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and I'll explain the
reason in a second--
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is this important result of
Lubotzky, Phillips, and Sarnak.
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From the late '80s.
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So their paper was
titled Ramanujan graphs.
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So they proved that
this conjecture is true.
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So the conjecture is true for
all d, such that d minus 1
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is a prime number.
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I should also remark
that the same result
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was proved independently by
Margulis at the same time.
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Their construction of this graph
is a specific Cayley graph.
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So they gave an explicit
construction of a Cayley graph,
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with the group being the
projective special linear
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group--
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PSL 2 q.
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So, some group-- and this
group actually comes up a lot.
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It's a group with lots of nice
pseudo-randomness properties.
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And to verify that,
the corresponding graph
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has the desired
eigenvalue properties,
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they had to invoke
some deep results
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from number theory
that were related
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to Ramanujan conjectures.
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So that's why they called
these graphs Ramanujan graphs.
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And that name stuck.
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So these papers, they
proved that these graphs
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exist for some special values
of d-- namely, when d minus 1
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is a prime.
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There was a later
generalization in the '90s,
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by Morgenstern, generalizing
such constructions
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showing that you can also take
d minus 1 to be a prime power.
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And really, that's
pretty much it.
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For all the other
values of d, it
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is open whether there exists
infinitely many d-regular
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Ramanujan graphs.
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In particular, for d equal
to 7, it is still open.
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Do there exist infinitely many
semi-regular Ramanujan graphs?
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No.
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What about a random graph?
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If I take a random
graph, what is
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the size of its second
largest eigenvalue?
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And there is a difficult
theorem of Friedman--
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I say difficult,
because the paper itself
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is more than 100 pages long--
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that if you take a fixed
d, then a random end
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vertex d-regular graph.
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So what does this mean?
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So the easiest way to explain
the random d-regular graph
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is that you look at the set of
all possible d-regular graphs
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on a fixed number of
vertices and you pick one
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uniformly at random.
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So random-- such graph
is almost Ramanujan,
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in the following sense--
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that the second largest
eigenvalue in absolute value
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is, at most, 2 root d minus
1 plus some small arrow
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little 1, where the
little 1 goes to 0
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as n goes to infinity.
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So in other words, this
constant cannot be improved,
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but this result doesn't tell
you that any of these graphs are
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Ramanujan.
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Experimental evidence
suggests that if you
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take, for fixed value of d--
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let's say d equals to
7 or d equals to 3--
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if you take a random
d-regular graph,
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then a specific percentage of
those graphs are Ramanujan.
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So the second largest
eigenvalue has
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some empirical
distribution, at least
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from computer experiments,
where some specific fraction--
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I don't remember
exactly, but let's
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say 40% of three
regular graphs--
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is expected in the
limit to be Ramanujan.
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So that appears to
be quite difficult.
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We have no idea how to even
approach such conjectures.
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There were some exciting
recent breakthroughs
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in the past few years concerning
a variant, a somewhat weakening
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of this problem--
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a bipartite analogue
of Ramanujan graphs.
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Now, in a bipartite graph--
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so all bipartite graphs
have the property
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that its eigenvalues,
its spectrum,
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is symmetric around 0.
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Its smallest
eigenvalue is minus d.
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So if you plot all
the eigenvalues,
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it's symmetric around 0.
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This is not a hard fact to see--
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I encourage you
to think about it.
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And that's because, if
you have an eigenvector--
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so it lifts
somewhere on the left
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and somewhere on the right--
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I can form another
eigenvector, which
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is obtained by flipping
the signs on one part.
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If the first eigenvector
has eigenvalue lambda,
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then the second one has
eigenvalue minus lambda.
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So the eigenvalues come
in symmetric pairs.
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So by definition, a
bipartite Ramanujan graph
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is one where it's
a bipartite graph
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and I only require that the
second largest eigenvalue
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is less than 2 root d minus 1.
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Everything's symmetric
around the origin.
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So this is by definition.
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If you start with
a Ramanujan graph,
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I can use it to create a
bipartite Ramanujan graph,
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because if I look
at this 2 lift--
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so where there's
this construction--
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this means if I start with
some graph, G-- so for example,
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if G is this graph
here, what I want to do
00:12:08.290 --> 00:12:11.320
is take two copies
of this graph,
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think about having them
on two sheets of paper,
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one on top of the other.
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And I draw all the
edges criss-crossed.
00:12:25.250 --> 00:12:28.100
So that's G cross K 2.
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This is G.
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You should convince
yourself that if G
00:12:32.750 --> 00:12:40.730
has eigenvalues lambda then
G cross K 2 has eigenvalues.
00:12:45.310 --> 00:12:48.070
The original spectrum,
as well, it's
00:12:48.070 --> 00:12:50.950
symmetric-- so it's negation.
00:12:50.950 --> 00:12:54.850
So if G is Ramanujan,
then G cross
00:12:54.850 --> 00:12:57.970
K 2 is a bipartite
Ramanujan graph.
00:12:57.970 --> 00:13:01.090
So it's a weaker concept--
if you have Ramanujan graphs,
00:13:01.090 --> 00:13:04.360
then you have bipartite
Ramanujan graphs-- but not
00:13:04.360 --> 00:13:04.920
in reverse.
00:13:08.970 --> 00:13:11.350
But, still a problem
of do there exist
00:13:11.350 --> 00:13:15.012
d-regular bipartite Ramanujan
graphs is still interesting.
00:13:15.012 --> 00:13:17.470
It's a somewhat weaker problem,
but it's still interesting.
00:13:17.470 --> 00:13:20.020
And there was a
major breakthrough
00:13:20.020 --> 00:13:32.070
a few years ago by Marcus
Spielman and Srivastava,
00:13:32.070 --> 00:13:34.860
showing that for
all fixed d, there
00:13:34.860 --> 00:13:42.370
exist infinitely many d-regular
bipartite Ramanujan graphs.
00:13:47.860 --> 00:13:50.950
And unlike the earlier
work of Lubotzky, Phillips,
00:13:50.950 --> 00:13:52.030
and Sarnak--
00:13:52.030 --> 00:13:55.480
which, the earlier work was
an explicit construction
00:13:55.480 --> 00:13:57.400
of a Cayley graph--
00:13:57.400 --> 00:14:01.650
this construction here is a
probabilistic construction.
00:14:01.650 --> 00:14:03.970
It uses some very
nice tools that they
00:14:03.970 --> 00:14:06.770
called interlacing families.
00:14:06.770 --> 00:14:09.460
So it showed,
probabilistically, using
00:14:09.460 --> 00:14:12.190
a very clever
randomized construction,
00:14:12.190 --> 00:14:14.060
that these graphs exist.
00:14:14.060 --> 00:14:17.770
So it's not just take a usual
d-regular random bipartite
00:14:17.770 --> 00:14:21.442
graph, but there's some clever
constructions of randomness.
00:14:24.400 --> 00:14:26.360
And this is more
or less the state
00:14:26.360 --> 00:14:30.680
of knowledge regarding the
existence of Ramanujan graphs.
00:14:30.680 --> 00:14:34.250
Again, the big open problem
is that there exists
00:14:34.250 --> 00:14:36.530
d-regular Ramanujan graphs.
00:14:36.530 --> 00:14:39.080
For every d, there
are infinitely many
00:14:39.080 --> 00:14:42.810
such Ramanujan graphs.
00:14:42.810 --> 00:14:43.786
Yeah.
00:14:43.786 --> 00:14:47.690
AUDIENCE: So in the
conception of G cross K 2,
00:14:47.690 --> 00:14:51.106
lambda 1 is equal to d?
00:14:51.106 --> 00:14:56.103
Or is equal to
original [INAUDIBLE]..
00:14:56.103 --> 00:14:57.520
YUFEI ZHAO: Right,
so the question
00:14:57.520 --> 00:15:01.390
is, if you start with
a d-regular graph
00:15:01.390 --> 00:15:09.180
and take this construction,
the spectrum has 1 d
00:15:09.180 --> 00:15:12.180
and it also has a minus d.
00:15:12.180 --> 00:15:14.550
If your graph is
bipartite, its spectrum
00:15:14.550 --> 00:15:16.390
is symmetric around the origin.
00:15:16.390 --> 00:15:19.180
So you always have
d and minus d.
00:15:19.180 --> 00:15:21.510
So a bipartite graph
can never be Ramanujan.
00:15:21.510 --> 00:15:24.400
But the definition of a
bipartite Ramanujan graph
00:15:24.400 --> 00:15:28.950
is just that I only require that
the remaining eigenvalues sit
00:15:28.950 --> 00:15:30.520
in that interval.
00:15:30.520 --> 00:15:33.790
I'm OK with having minus d here.
00:15:33.790 --> 00:15:36.940
So that's by definition of
a bipartite Ramanujan graph.
00:15:39.900 --> 00:15:41.131
Any more questions?
00:15:44.220 --> 00:15:49.880
All right, so combining
a Alon-Boppana bound
00:15:49.880 --> 00:15:52.970
and both the existence
of Ramanujan graphs
00:15:52.970 --> 00:15:54.710
and also Friedman's
difficult result
00:15:54.710 --> 00:15:57.530
that random graph
is almost Ramanujan,
00:15:57.530 --> 00:16:01.640
we see that this
2 root d minus 1--
00:16:01.640 --> 00:16:03.050
that number there is optimal.
00:16:06.520 --> 00:16:09.300
So that's the extent in
which a d-regular graph
00:16:09.300 --> 00:16:10.580
can be pseudo-random.
00:16:14.580 --> 00:16:16.330
Now, the rest of
this lecture, I want
00:16:16.330 --> 00:16:18.670
to move onto a somewhat
different topic,
00:16:18.670 --> 00:16:22.210
but still concerning sparse
pseudo-random graphs.
00:16:22.210 --> 00:16:24.730
Basically, I want to tell you
what I did for my PhD thesis.
00:16:28.140 --> 00:16:33.820
So, so far we've been talking
about pseudo-random graphs,
00:16:33.820 --> 00:16:36.640
but let's combine it with the
topic in the previous chapter--
00:16:36.640 --> 00:16:39.940
namely, similarities
regularity lemma.
00:16:39.940 --> 00:16:44.590
And we can ask, can we
apply the regularity method
00:16:44.590 --> 00:16:46.570
to sparse graphs?
00:16:49.260 --> 00:16:51.360
So when we talk about
similarities regularity,
00:16:51.360 --> 00:16:54.180
I kept emphasizing that it's
really about dense graphs,
00:16:54.180 --> 00:16:56.400
because there are these
error terms which are little
00:16:56.400 --> 00:16:57.240
and squared.
00:16:57.240 --> 00:16:58.740
And if your graph
is already sparse,
00:16:58.740 --> 00:17:01.190
that error term
eats up everything.
00:17:01.190 --> 00:17:05.260
So for sparse graphs, you
need to be extra careful.
00:17:05.260 --> 00:17:08.530
So I want to explore the
idea of a sparse regularity.
00:17:08.530 --> 00:17:11.380
And here, sparse
just means not dense.
00:17:11.380 --> 00:17:20.210
So sparse means x
density, little 1.
00:17:20.210 --> 00:17:24.000
So, the opposite of dense.
00:17:24.000 --> 00:17:26.597
We saw the triangle
removal [INAUDIBLE]..
00:17:35.680 --> 00:17:37.290
So, let me remind
you the statement.
00:17:37.290 --> 00:17:46.320
It says that for every epsilon,
there exists some delta, such
00:17:46.320 --> 00:18:01.720
that if G has a small
number of triangles,
00:18:01.720 --> 00:18:16.880
then G can be made
triangle-free by removing
00:18:16.880 --> 00:18:17.950
a small number of edges.
00:18:31.250 --> 00:18:35.030
I would like to state a sparse
version of this theorem that
00:18:35.030 --> 00:18:39.650
works for graphs where I'm
looking at sub constant x
00:18:39.650 --> 00:18:42.222
densities.
00:18:42.222 --> 00:18:43.930
So roughly, this is
how it's going to go.
00:18:46.900 --> 00:18:50.610
I'm going to put in
these extra p factors.
00:18:50.610 --> 00:18:52.360
And you should think
of p as some quantity
00:18:52.360 --> 00:18:54.960
that goes to 0 with n.
00:19:00.240 --> 00:19:04.170
So, think of p as
the general scale.
00:19:04.170 --> 00:19:07.380
So that's the x density
scale we're thinking of.
00:19:07.380 --> 00:19:10.260
I would like to
say that if G has
00:19:10.260 --> 00:19:13.800
less than that many
triangles, then
00:19:13.800 --> 00:19:17.985
G can be made free by deleting
a small number of edges.
00:19:17.985 --> 00:19:19.740
But what does small mean here?
00:19:19.740 --> 00:19:22.620
Small should be relative
to the scale of x
00:19:22.620 --> 00:19:24.550
densities you're looking at.
00:19:24.550 --> 00:19:30.060
So in this case, we should add
an extra factor of p over here.
00:19:30.060 --> 00:19:32.370
So that's the kind of
statement I would like,
00:19:32.370 --> 00:19:35.940
but of course, this is
too good to be true,
00:19:35.940 --> 00:19:37.980
because we haven't
really modified anything.
00:19:37.980 --> 00:19:41.370
If you read the statement,
it's just completely false.
00:19:41.370 --> 00:19:45.780
So I would like adding some
conditions, some hypotheses,
00:19:45.780 --> 00:19:49.680
that would make such
a statement true.
00:19:49.680 --> 00:19:52.630
And hypothesis is going to
be roughly along those lines.
00:19:52.630 --> 00:19:55.440
So I'm going to call this a
meta-theorem, because I won't
00:19:55.440 --> 00:19:58.320
state the hypothesis precisely.
00:19:58.320 --> 00:20:00.510
But roughly, it will
be along the lines
00:20:00.510 --> 00:20:08.440
that, if gamma is
some, say, sufficiently
00:20:08.440 --> 00:20:29.350
pseudo-random graph and
vertices and x density, p, and G
00:20:29.350 --> 00:20:38.030
is a subgraph of
gamma, then I want
00:20:38.030 --> 00:20:42.830
to say that G has this triangle
removal property, relatively
00:20:42.830 --> 00:20:45.700
inside gamma.
00:20:45.700 --> 00:20:47.630
And this is true.
00:20:47.630 --> 00:20:49.330
Well, it is true
if you're putting
00:20:49.330 --> 00:20:52.870
the appropriate, sufficiently
pseudo-random condition.
00:20:52.870 --> 00:20:54.035
So I'm leaving here--
00:20:54.035 --> 00:20:55.910
I'll tell you more later
what this should be.
00:20:58.560 --> 00:21:00.750
So this is a kind of
statement that I would like.
00:21:00.750 --> 00:21:05.160
So a sparse extension of
the triangle removal lemma
00:21:05.160 --> 00:21:08.580
says that, if you have a
sufficiently pseudo-random
00:21:08.580 --> 00:21:12.090
host, or you think of this
gamma as a host graph,
00:21:12.090 --> 00:21:17.760
then inside that host, relative
to the density of this host,
00:21:17.760 --> 00:21:21.510
everything should
behave nicely, as you
00:21:21.510 --> 00:21:25.040
would expect in the dense case.
00:21:25.040 --> 00:21:29.420
The dense case is also a
special case of the sparse case,
00:21:29.420 --> 00:21:34.130
because if we took gamma
to be the complete graph--
00:21:34.130 --> 00:21:37.560
which is also pseudo-random,
it's everything--
00:21:37.560 --> 00:21:41.730
it's uniform-- then
this is also true.
00:21:41.730 --> 00:21:44.060
And that's triangle removal
among the dense case,
00:21:44.060 --> 00:21:48.520
but we want this
sparse extension.
00:21:48.520 --> 00:21:49.235
Question.
00:21:49.235 --> 00:21:53.690
AUDIENCE: Where does the
c come into [INAUDIBLE]??
00:21:53.690 --> 00:21:57.570
YUFEI ZHAO: Where
does p come in to--
00:21:57.570 --> 00:22:01.900
so p here is the edge
density of gamma.
00:22:01.900 --> 00:22:04.470
AUDIENCE: [INAUDIBLE]
00:22:04.470 --> 00:22:05.750
YUFEI ZHAO: Yeah.
00:22:05.750 --> 00:22:08.360
So again, I'm not really
stating this so precisely,
00:22:08.360 --> 00:22:12.490
but you should think
of p as something
00:22:12.490 --> 00:22:16.240
that could decay with n--
00:22:16.240 --> 00:22:20.440
not too quickly, but decay
at n to the minus some
00:22:20.440 --> 00:22:23.380
small constant.
00:22:23.380 --> 00:22:24.190
Yeah.
00:22:24.190 --> 00:22:27.107
AUDIENCE: Delta here
doesn't depend on gamma?
00:22:27.107 --> 00:22:27.940
YUFEI ZHAO: Correct.
00:22:27.940 --> 00:22:30.820
So the question is, what
does delta depend on?
00:22:30.820 --> 00:22:36.250
So here, delta depends
on only epsilon.
00:22:36.250 --> 00:22:38.830
And in fact, what
we would like--
00:22:38.830 --> 00:22:41.180
and this will indeed
be basically true--
00:22:41.180 --> 00:22:44.140
is that delta is more
or less the same delta
00:22:44.140 --> 00:22:46.383
from the original
triangle removal lemma.
00:22:49.624 --> 00:22:51.476
Yeah.
00:22:51.476 --> 00:22:53.660
AUDIENCE: If G is
any graph, what's
00:22:53.660 --> 00:22:57.772
stopping you from
making it a subgraph
00:22:57.772 --> 00:23:01.750
of some large [INAUDIBLE]?
00:23:01.750 --> 00:23:04.690
YUFEI ZHAO: So the question
is, if G is some arbitrary
00:23:04.690 --> 00:23:06.890
graph, what's to
stop you from making
00:23:06.890 --> 00:23:10.030
it a subgraph of a large
pseudo-random graph?
00:23:10.030 --> 00:23:11.710
And there's a great question.
00:23:11.710 --> 00:23:13.390
If I give you a
graph, G, can you
00:23:13.390 --> 00:23:16.190
test whether G satisfies
the hypothesis?
00:23:16.190 --> 00:23:18.190
Because the conclusion
doesn't depend on gamma.
00:23:18.190 --> 00:23:21.040
The conclusion is only
on G, but the hypothesis
00:23:21.040 --> 00:23:23.200
requires us to gamma.
00:23:23.200 --> 00:23:25.673
And so my two answers
to that is, one,
00:23:25.673 --> 00:23:27.340
you cannot always
embed it in the gamma.
00:23:33.037 --> 00:23:34.829
I guess the easier
answer is the conclusion
00:23:34.829 --> 00:23:38.273
is false with other hypothesis.
00:23:38.273 --> 00:23:40.190
So you cannot always
embed it in such a gamma.
00:23:40.190 --> 00:23:43.162
But it is somewhat
difficult to test.
00:23:43.162 --> 00:23:45.120
I don't know a good way
to test whether a given
00:23:45.120 --> 00:23:46.850
G lies in such a gamma.
00:23:46.850 --> 00:23:48.600
I will motivate this
theorem in a second--
00:23:48.600 --> 00:23:52.160
why we care about
results of this form.
00:23:52.160 --> 00:23:52.735
Yes.
00:23:52.735 --> 00:23:57.020
AUDIENCE: Don't all sufficiently
large pseudo-random graphs--
00:23:57.020 --> 00:23:59.953
say, with respect to the
number of vertices of G--
00:23:59.953 --> 00:24:03.780
contain copies of every G?
00:24:03.780 --> 00:24:05.610
YUFEI ZHAO: So the
question is, if you
00:24:05.610 --> 00:24:09.410
start with a sufficiently
large pseudo-random gamma,
00:24:09.410 --> 00:24:10.950
does it contain every copy of G?
00:24:10.950 --> 00:24:14.190
And the answer is no, because G
has the same number of vertices
00:24:14.190 --> 00:24:16.530
as gamma.
00:24:16.530 --> 00:24:18.330
A sufficiently
pseudo-random-- again,
00:24:18.330 --> 00:24:20.622
I haven't told you what
sufficiently pseudo-random even
00:24:20.622 --> 00:24:21.280
means yet.
00:24:21.280 --> 00:24:25.890
But you should think of it as
controlling small patterns.
00:24:25.890 --> 00:24:28.200
But here, G is a
much larger graph.
00:24:28.200 --> 00:24:30.700
It's the same size,
it's just maybe,
00:24:30.700 --> 00:24:33.240
let's say, half of the edges.
00:24:33.240 --> 00:24:36.990
So what you should think about
is starting with gamma being,
00:24:36.990 --> 00:24:38.400
let's say, a random graph.
00:24:38.400 --> 00:24:41.310
And I delete
adversarially, let's say,
00:24:41.310 --> 00:24:43.720
half of the edges of gamma.
00:24:43.720 --> 00:24:47.220
And you get G.
00:24:47.220 --> 00:24:48.280
So let me go on.
00:24:48.280 --> 00:24:51.480
And please ask more questions.
00:24:51.480 --> 00:24:53.460
I won't really prove
anything today,
00:24:53.460 --> 00:24:55.290
but it's really
meant to give you
00:24:55.290 --> 00:25:00.090
an idea of what this
line of work is about.
00:25:00.090 --> 00:25:05.000
And I also want to motivate
it by explaining why we care
00:25:05.000 --> 00:25:06.590
about these kind of theorems.
00:25:06.590 --> 00:25:08.995
So first observation
is that it is not
00:25:08.995 --> 00:25:11.120
true with all the hypothesis--
hopefully all of you
00:25:11.120 --> 00:25:13.850
see this as obviously
too good to be true.
00:25:13.850 --> 00:25:18.530
But will also see some
specific examples.
00:25:18.530 --> 00:25:19.840
Here's a specific example.
00:25:19.840 --> 00:25:27.400
So this is not true
without this gamma.
00:25:27.400 --> 00:25:30.650
So for example, you
can have this graph, G.
00:25:30.650 --> 00:25:32.710
And we already saw
this construction that
00:25:32.710 --> 00:25:37.540
came from Behren's construction,
where you have n vertices
00:25:37.540 --> 00:25:49.780
and n to the 2 minus little 1
edges, where every edge belongs
00:25:49.780 --> 00:25:51.130
to exactly one triangle.
00:25:57.400 --> 00:26:00.060
If you plug in this
graph into this theorem,
00:26:00.060 --> 00:26:01.800
with all the yellow stuff--
00:26:01.800 --> 00:26:03.400
if you add in this p--
00:26:03.400 --> 00:26:04.660
you see it's false.
00:26:04.660 --> 00:26:08.125
You just cannot remove--
00:26:08.125 --> 00:26:08.625
anyway.
00:26:11.860 --> 00:26:17.010
In what context can we expect
such a sparse triangle removal
00:26:17.010 --> 00:26:18.500
lemma to be true?
00:26:18.500 --> 00:26:20.470
One setting for
which it is true--
00:26:20.470 --> 00:26:23.500
and this was a result that was
proved about 10 years ago--
00:26:23.500 --> 00:26:26.730
is that if your gamma
is a truly random graph.
00:26:26.730 --> 00:26:35.170
So this is true
for a random gamma
00:26:35.170 --> 00:26:42.140
if p is sufficiently large
and roughly it's at least--
00:26:42.140 --> 00:26:45.830
so there's some constant
such that if p is at least c
00:26:45.830 --> 00:26:47.413
over [INAUDIBLE],,
then it is true.
00:26:47.413 --> 00:26:49.205
So this is the result
of Conlon and Gowers.
00:26:56.686 --> 00:26:57.186
Yeah.
00:26:57.186 --> 00:27:00.540
AUDIENCE: Is this random
in the Erdos-Rényi sense?
00:27:00.540 --> 00:27:03.712
YUFEI ZHAO: So this is random
in the Erdos-Rényi sense.
00:27:03.712 --> 00:27:04.920
So, Erdos-Rényi random graph.
00:27:07.610 --> 00:27:09.710
But this is not the
main motivating reason
00:27:09.710 --> 00:27:12.170
why I would like to talk
about this technique.
00:27:12.170 --> 00:27:17.210
The main motivating example
is the Green-Tao theorem.
00:27:25.180 --> 00:27:29.090
So I remind you that
the Green-Tao theorem
00:27:29.090 --> 00:27:32.840
says that the primes contain
arbitrarily long arithmetic
00:27:32.840 --> 00:27:33.848
progressions.
00:27:52.280 --> 00:27:55.020
So the Green-Tao theorem
is, in some sense,
00:27:55.020 --> 00:27:57.300
an extension of
Szemeredi's theorem,
00:27:57.300 --> 00:27:59.072
but a sparse extension.
00:27:59.072 --> 00:28:00.780
Szemeredi's theorem
tells you that if you
00:28:00.780 --> 00:28:04.090
have a positive density
subset of the integers, then
00:28:04.090 --> 00:28:07.670
it contains long
arithmetic progressions.
00:28:07.670 --> 00:28:09.200
But here, the primes--
00:28:09.200 --> 00:28:13.690
we know from prime
number theorem
00:28:13.690 --> 00:28:19.750
that the density of
the primes up to n
00:28:19.750 --> 00:28:24.580
decays, like 1 over log n.
00:28:24.580 --> 00:28:27.490
So it's a sparse
set, but we would
00:28:27.490 --> 00:28:30.923
like to know that it has
all of these patterns.
00:28:30.923 --> 00:28:33.340
It turns out the primes are,
in some sense, pseudo-random.
00:28:33.340 --> 00:28:35.850
But that's a difficult
result to prove.
00:28:35.850 --> 00:28:38.830
And that was proved after
Green-Tao proved their initial
00:28:38.830 --> 00:28:40.032
theorem--
00:28:40.032 --> 00:28:44.590
so, by later works of Green
and Tao and also Ziegler.
00:28:44.590 --> 00:28:46.360
But the original strategy--
00:28:46.360 --> 00:28:49.350
and also the later strategy for
the stronger result, as well.
00:28:49.350 --> 00:28:51.760
But the strategy for the
Green-Tao theorem is this--
00:28:51.760 --> 00:28:56.020
you start with the primes
and you embed the primes
00:28:56.020 --> 00:28:57.370
in a somewhat larger set.
00:29:02.360 --> 00:29:05.210
You start with the
primes and you embed it
00:29:05.210 --> 00:29:09.260
in a somewhat larger set,
which we'll call, informally,
00:29:09.260 --> 00:29:12.580
pseudoprimes.
00:29:12.580 --> 00:29:17.014
And these m, roughly
speaking, numbers
00:29:17.014 --> 00:29:21.195
with no small prime divisors.
00:29:34.000 --> 00:29:37.360
Because these numbers
are somewhat smoother
00:29:37.360 --> 00:29:39.160
compared to the
primes, they're easier
00:29:39.160 --> 00:29:42.830
to analyze by analytic
number theory methods,
00:29:42.830 --> 00:29:44.950
especially coming
from sieve theory.
00:29:44.950 --> 00:29:49.390
And it is easier, although
still highly nontrivial,
00:29:49.390 --> 00:29:52.630
to show that these pseudoprimes
are, in some sense,
00:29:52.630 --> 00:29:53.860
pseudo-random.
00:30:02.020 --> 00:30:04.490
And that's the kind
of pseudo-random host
00:30:04.490 --> 00:30:08.510
that corresponds with
the gamma over there.
00:30:08.510 --> 00:30:11.560
So the Green-Tao strategy
is to start with the primes,
00:30:11.560 --> 00:30:14.980
build a slightly larger
set so that the prime sit
00:30:14.980 --> 00:30:18.460
inside the pseudoprimes in
a relatively dense manner.
00:30:26.480 --> 00:30:30.930
So it has high relative density.
00:30:30.930 --> 00:30:34.610
And then, if you had this
kind of strategy for a sparse
00:30:34.610 --> 00:30:36.140
triangle removal lemma--
00:30:36.140 --> 00:30:40.010
but imagine you also had it
for various other extensions
00:30:40.010 --> 00:30:43.160
of sparse hypergraph removal
lemma, which allows you
00:30:43.160 --> 00:30:45.050
to prove Szemeredi's theorem.
00:30:45.050 --> 00:30:48.000
And now you can use
it in that setting.
00:30:48.000 --> 00:30:52.220
Then you can prove Szemeredi's
theorem in the primes.
00:30:55.060 --> 00:30:57.520
That's the theorem and
that's the approach.
00:30:57.520 --> 00:30:59.740
And that's one of the
reasons, at least for me,
00:30:59.740 --> 00:31:04.090
why something like a sparse
triangle removal lemma
00:31:04.090 --> 00:31:07.810
plays a central role in
these kind of problems.
00:31:12.500 --> 00:31:15.790
So I want to say
more about how you
00:31:15.790 --> 00:31:18.040
might go about proving
this type of result
00:31:18.040 --> 00:31:23.030
and also what pseudo-random
graph means over here.
00:31:23.030 --> 00:31:28.150
So, remember the strategy for
proving the triangle removal
00:31:28.150 --> 00:31:28.840
lemma.
00:31:28.840 --> 00:31:32.580
And of course, all of you guys
are working on this problem set
00:31:32.580 --> 00:31:35.470
and so the method of
regularity hopefully
00:31:35.470 --> 00:31:39.140
should be very familiar to
you by the end of this week.
00:31:39.140 --> 00:31:43.510
But let me remind you that
there are three main steps, one
00:31:43.510 --> 00:31:49.650
being to partition your graph
using the regularity lemma.
00:31:49.650 --> 00:31:51.640
The second one, to clean.
00:31:51.640 --> 00:31:53.230
And the third one, the count.
00:31:56.350 --> 00:32:05.360
And I want to explain where the
sparse regularity method fails.
00:32:05.360 --> 00:32:07.777
So you can try to do
everything the same and then--
00:32:07.777 --> 00:32:09.860
so what happens if you try
to do all these things?
00:32:13.790 --> 00:32:19.620
So first, let's talk about
sparse regularity lemma.
00:32:30.710 --> 00:32:32.680
So let me remind
you-- previously,
00:32:32.680 --> 00:32:43.230
we said that a pair of
vertices is epsilon regular
00:32:43.230 --> 00:32:52.150
if, for every subset
U of A and W of B--
00:32:52.150 --> 00:32:53.020
neither too small.
00:32:56.800 --> 00:33:08.950
So if neither are
too small, one has
00:33:08.950 --> 00:33:13.600
that the number of
edges between U and W
00:33:13.600 --> 00:33:19.000
differs from what
you would expect.
00:33:19.000 --> 00:33:25.870
So the x density between U and
W is close to what you expect,
00:33:25.870 --> 00:33:30.450
which is the ordinal edge
density between A and B.
00:33:30.450 --> 00:33:32.930
So they differ by no
more than epsilon.
00:33:36.800 --> 00:33:39.470
So this should be a
familiar definition.
00:33:39.470 --> 00:33:41.690
What we would like
is to modify it
00:33:41.690 --> 00:33:43.720
to work for the sparse setting.
00:33:43.720 --> 00:33:46.730
And for that, I'm going to
add in an extra p factor.
00:33:46.730 --> 00:33:51.350
So I'm going to say
epsilon, p regular.
00:33:51.350 --> 00:33:54.260
Well, this condition here--
now, oh, the densities
00:33:54.260 --> 00:33:56.900
are on the scale of p.
00:33:56.900 --> 00:33:58.980
Which goes to 0 as
n goes to infinity.
00:33:58.980 --> 00:34:01.040
So in what does the
property compare them?
00:34:01.040 --> 00:34:03.950
I should add an
extra factor of p
00:34:03.950 --> 00:34:06.690
to put everything
on the right scale.
00:34:06.690 --> 00:34:07.880
Otherwise, this is too weak.
00:34:16.840 --> 00:34:19.460
And given this
definition here, we
00:34:19.460 --> 00:34:29.580
can say that a
partition of vertices
00:34:29.580 --> 00:34:41.610
is epsilon regular if all part
but at most epsilon fraction
00:34:41.610 --> 00:34:47.489
pairs is epsilon regular--
00:34:47.489 --> 00:34:52.219
so an equitable partition.
00:34:52.219 --> 00:34:54.510
And I would modify it
to the sparse setting
00:34:54.510 --> 00:34:57.990
by changing the appropriate
notion of regular
00:34:57.990 --> 00:35:03.600
to the sparse version, where
I'm looking at scales of p.
00:35:03.600 --> 00:35:06.367
I still require at
most epsilon fraction--
00:35:06.367 --> 00:35:07.200
that stays the same.
00:35:07.200 --> 00:35:08.908
That's not affected
by the density scale.
00:35:11.540 --> 00:35:23.070
Previously, we had the
irregularity lemma,
00:35:23.070 --> 00:35:27.340
which said that
for every epsilon,
00:35:27.340 --> 00:35:40.390
there exists some M such
that every graph has
00:35:40.390 --> 00:35:49.330
an epsilon regular partition
into at most M parts.
00:35:54.030 --> 00:36:01.750
And the sparse version
would say that if your graph
00:36:01.750 --> 00:36:09.325
has x density at most p--
00:36:09.325 --> 00:36:11.780
and here, all of these
constants are negotiable.
00:36:11.780 --> 00:36:14.570
So when I say p, I really
could mean 100 times p.
00:36:14.570 --> 00:36:16.400
You just change these constants.
00:36:16.400 --> 00:36:22.040
So if it's most p, then it has
an epsilon, p regular partition
00:36:22.040 --> 00:36:24.320
into at most m parts.
00:36:24.320 --> 00:36:26.600
Here, m depends only on epsilon.
00:36:29.730 --> 00:36:34.300
So previously, I wrote down the
sparse triangle removal lemma.
00:36:34.300 --> 00:36:37.740
And I wrote down the
statement and it was false--
00:36:37.740 --> 00:36:41.290
with all the additional
hypotheses, it was false.
00:36:41.290 --> 00:36:44.190
It turns out that this
is actually true--
00:36:44.190 --> 00:36:47.010
the version of the
sparse regularity lemma,
00:36:47.010 --> 00:36:49.950
which sounds almost too
good to be true, initially.
00:36:49.950 --> 00:36:56.130
We are adding in a
whole lot of sparsity
00:36:56.130 --> 00:37:00.450
and sparsity seems to be
more difficult to deal with.
00:37:00.450 --> 00:37:03.750
And the reason why I
think sparsity is harder
00:37:03.750 --> 00:37:06.810
to deal with is
that, in some sense,
00:37:06.810 --> 00:37:09.030
there are a lot
more sparse graphs
00:37:09.030 --> 00:37:11.500
than there are dense graphs.
00:37:11.500 --> 00:37:16.050
So let me pause for a
second and explain that.
00:37:16.050 --> 00:37:18.900
It is not true
that, in some sense,
00:37:18.900 --> 00:37:21.330
there are more sparse graphs.
00:37:21.330 --> 00:37:23.280
Because if you just count--
00:37:23.280 --> 00:37:25.870
once you have sparser things,
there are fewer of them.
00:37:25.870 --> 00:37:29.250
But I mean in terms of
the actual complexity
00:37:29.250 --> 00:37:31.200
of the structures
that can come up.
00:37:31.200 --> 00:37:33.600
When you have sparser
objects, there's
00:37:33.600 --> 00:37:35.940
a lot more that can happen.
00:37:35.940 --> 00:37:39.590
In dense objects,
Szemeredi's regularity lemma
00:37:39.590 --> 00:37:42.830
tells us, in some sense,
that the amount of complexity
00:37:42.830 --> 00:37:45.260
in the graph is bounded.
00:37:45.260 --> 00:37:49.270
But that's not the
case for sparse graphs.
00:37:49.270 --> 00:37:50.770
In any case, we
still have some kind
00:37:50.770 --> 00:37:53.400
of sparse regularity lemma.
00:37:53.400 --> 00:37:55.710
And this version
here, as written,
00:37:55.710 --> 00:37:58.770
is literally true if you have
the appropriate definitions--
00:37:58.770 --> 00:38:02.240
and more or less, we have
those definitions up there.
00:38:02.240 --> 00:38:06.580
But I want to say
that it's misleading.
00:38:06.580 --> 00:38:09.683
This is true, but misleading.
00:38:15.250 --> 00:38:17.320
And the reason why
it is misleading
00:38:17.320 --> 00:38:19.420
is that, in a sparse
graph, you can
00:38:19.420 --> 00:38:23.570
have lots of
intricate structures
00:38:23.570 --> 00:38:26.950
that are hidden in
your irregular parts.
00:38:26.950 --> 00:38:46.480
It could be that most edges are
inside irregular pairs, which
00:38:46.480 --> 00:38:48.970
would make the irregularity
lemma a somewhat
00:38:48.970 --> 00:38:53.210
useless statement, because
when you do the cleaning step,
00:38:53.210 --> 00:38:55.010
you delete all of your edges.
00:38:55.010 --> 00:38:56.010
And you don't want that.
00:38:59.800 --> 00:39:01.690
But in any case, it is true--
00:39:01.690 --> 00:39:03.480
and I'll comment on
the proof in a second.
00:39:03.480 --> 00:39:06.390
But the way I want you to think
about the sparse regularity
00:39:06.390 --> 00:39:11.620
lemma is that it
should work when--
00:39:11.620 --> 00:39:14.680
so before jumping to
that, a specific example
00:39:14.680 --> 00:39:17.330
where this happens
is, for example,
00:39:17.330 --> 00:39:25.690
if your graph G is a click on a
sublinear fraction of vertices.
00:39:25.690 --> 00:39:28.100
Somehow, you might
care about that click.
00:39:28.100 --> 00:39:30.640
So that's a pretty important
object in the graph.
00:39:30.640 --> 00:39:33.250
But when you do the sparse
regularity partition,
00:39:33.250 --> 00:39:35.380
it could be that the
entire click is hidden
00:39:35.380 --> 00:39:39.450
inside an irregular part.
00:39:39.450 --> 00:39:40.720
And you just don't see it--
00:39:40.720 --> 00:39:41.950
that information gets lost.
00:39:45.230 --> 00:39:48.260
The proper way to think about
the sparse regularity lemma
00:39:48.260 --> 00:39:51.800
is to think about
graphs, G, that satisfy
00:39:51.800 --> 00:39:54.720
some additional hypotheses.
00:39:54.720 --> 00:40:07.810
So in practice, g is assumed to
satisfy some upper regularity
00:40:07.810 --> 00:40:08.778
condition.
00:40:17.570 --> 00:40:19.880
And an example of
such an hypothesis
00:40:19.880 --> 00:40:25.610
is something called
no dense spots,
00:40:25.610 --> 00:40:28.580
meaning that it doesn't have
a really dense component,
00:40:28.580 --> 00:40:33.220
like in the case of a click on
a very small number of vertices.
00:40:33.220 --> 00:40:36.240
So no dense spots--
00:40:36.240 --> 00:40:41.140
one definition could be
that there exists some eta--
00:40:41.140 --> 00:40:43.990
and here, just as in
quasi-random graphs,
00:40:43.990 --> 00:40:46.030
I'm thinking of
sequences going to 0.
00:40:46.030 --> 00:40:47.890
So there exists
eta sequence going
00:40:47.890 --> 00:40:58.250
to 0 and a constant,
c, such that for all
00:40:58.250 --> 00:41:01.760
set x in the graph--
00:41:01.760 --> 00:41:11.470
let's say X and Y. If X and Y
have size at least eta fraction
00:41:11.470 --> 00:41:18.380
of V, then the density
between X and Y
00:41:18.380 --> 00:41:22.760
is bounded by at most
a constant factor,
00:41:22.760 --> 00:41:27.180
compared to the overall density,
p, that we're looking at.
00:41:27.180 --> 00:41:30.350
So in other words, no
small piece of the graph
00:41:30.350 --> 00:41:31.910
has too many edges.
00:41:41.440 --> 00:41:50.980
And with that notion
of the no dense spots,
00:41:50.980 --> 00:41:55.090
we can now prove the
sparse regularity lemma
00:41:55.090 --> 00:41:58.090
under that additional
hypothesis.
00:41:58.090 --> 00:42:00.700
And basically, the
proof is the same
00:42:00.700 --> 00:42:04.030
as the usual
semi-regularity lemma proof
00:42:04.030 --> 00:42:06.420
that we saw a few weeks ago.
00:42:09.360 --> 00:42:16.746
So if you have proof
of sparse regularity
00:42:16.746 --> 00:42:19.540
with no dense spots--
00:42:23.530 --> 00:42:24.210
hypothesis.
00:42:27.150 --> 00:42:32.115
OK, so I claim this as the
same proof as Szemeredi's
00:42:32.115 --> 00:42:34.230
irregularity lemma.
00:42:34.230 --> 00:42:41.015
And the reason is that in the
energy increment argument,
00:42:41.015 --> 00:42:42.140
you do everything the same.
00:42:42.140 --> 00:42:43.807
You do partitioning
if it's not regular.
00:42:43.807 --> 00:42:47.410
You refine and you keep going.
00:42:47.410 --> 00:42:50.410
In the energy
increment argument,
00:42:50.410 --> 00:42:52.690
one key property we
used was that the energy
00:42:52.690 --> 00:42:55.000
was bounded between 0 and 1.
00:42:55.000 --> 00:42:59.820
And every time, you went
up by epsilon to the fifth.
00:42:59.820 --> 00:43:04.110
And now the energy
increment argument--
00:43:04.110 --> 00:43:10.890
that each step,
the energy goes up
00:43:10.890 --> 00:43:16.140
by something which
is like epsilon,
00:43:16.140 --> 00:43:21.910
let's say, to the
fifth and p squared.
00:43:21.910 --> 00:43:25.860
The energy is some kind
of mean square density,
00:43:25.860 --> 00:43:28.070
so this p squared
should play a role.
00:43:34.030 --> 00:43:37.630
So if you only knew that,
then the number of iterations
00:43:37.630 --> 00:43:39.850
might depend on p--
00:43:39.850 --> 00:43:41.730
it might depend on n.
00:43:41.730 --> 00:43:45.360
So, not a constant-- and
that would be an issue.
00:43:45.360 --> 00:43:50.150
However, if you have
no dense spots--
00:43:50.150 --> 00:43:57.190
so, because no dense spots--
00:43:57.190 --> 00:44:08.080
the final energy, I claim,
is, at most, something
00:44:08.080 --> 00:44:11.350
like C squared p squared.
00:44:11.350 --> 00:44:15.430
Maybe some small error,
because of all the epsilons
00:44:15.430 --> 00:44:19.680
flowing around, but
that's the final energy.
00:44:19.680 --> 00:44:23.850
So you still have a
bounded number of steps.
00:44:28.730 --> 00:44:33.650
So the bound only
depends on epsilon.
00:44:33.650 --> 00:44:35.970
So the entire proof
runs through just fine.
00:44:40.500 --> 00:44:43.520
OK, so having the
right hypothesis helps.
00:44:43.520 --> 00:44:45.830
But then I said, the
more general version
00:44:45.830 --> 00:44:49.350
without the hypothesis
is still true.
00:44:49.350 --> 00:44:51.220
So how come that is the case?
00:44:51.220 --> 00:44:53.470
Because if you do this proof,
you run into the issue--
00:44:53.470 --> 00:44:57.680
you cannot control the
number of iterations.
00:44:57.680 --> 00:45:02.150
So here's a trick introduced
by Alex Scott, who came up
00:45:02.150 --> 00:45:03.400
with that version there.
00:45:06.740 --> 00:45:13.280
So this is a nice trick, which
is that, instead of using
00:45:13.280 --> 00:45:14.170
x squared--
00:45:14.170 --> 00:45:18.640
the function as energy--
00:45:18.640 --> 00:45:22.780
let's consider a somewhat
different function.
00:45:22.780 --> 00:45:29.220
So the function I want to use
is fe of x which is initially
00:45:29.220 --> 00:45:32.220
quadratic-- so,
initially x squared--
00:45:32.220 --> 00:45:35.850
but up to a specific point.
00:45:35.850 --> 00:45:36.480
Let's say 2.
00:45:40.120 --> 00:45:42.520
And then after this
point, I make it linear.
00:45:51.560 --> 00:45:54.030
So that's the function
I'm going to take.
00:45:54.030 --> 00:45:56.940
Now, this function has a
couple of nice properties.
00:45:56.940 --> 00:46:03.110
One is that you also have this
boosting, this energy increment
00:46:03.110 --> 00:46:08.780
step, because for all
random variables, x--
00:46:12.450 --> 00:46:14.780
so x is a non-negative
random variable.
00:46:14.780 --> 00:46:16.970
So think of this as edge
densities between parts
00:46:16.970 --> 00:46:18.770
on the refinement.
00:46:18.770 --> 00:46:26.100
If the mean of x is,
at most, 1, then,
00:46:26.100 --> 00:46:35.250
if you look at this
energy, it increases
00:46:35.250 --> 00:46:41.475
if x has a large variance.
00:46:44.810 --> 00:46:48.710
Previously, when we used fe
as square, this was true.
00:46:48.710 --> 00:46:50.390
So this is true
with equal to 1--
00:46:50.390 --> 00:46:52.610
in fact, that's the
definition of variance.
00:46:52.610 --> 00:46:55.700
But this inequality is also
true for this function, fe--
00:46:55.700 --> 00:46:59.210
so that when you do the
irregularity breaking,
00:46:59.210 --> 00:47:01.610
if you have irregular
parts, then you
00:47:01.610 --> 00:47:05.060
have some variance in
the edge densities.
00:47:05.060 --> 00:47:08.390
So you would get
an energy boost.
00:47:08.390 --> 00:47:11.900
But the other thing
is that we are
00:47:11.900 --> 00:47:16.880
no longer worried about
the final energy being
00:47:16.880 --> 00:47:20.060
much higher than the individual
potential contributions.
00:47:20.060 --> 00:47:28.360
Because, if you end up having
lots of high density pieces,
00:47:28.360 --> 00:47:31.150
they would contribute a lot.
00:47:31.150 --> 00:47:44.180
So, in other words,
the expectation
00:47:44.180 --> 00:47:49.630
for the second thing is
that the expectation of fe
00:47:49.630 --> 00:48:03.410
is upper-bounded by, let's say,
4 times the expectation of x.
00:48:03.410 --> 00:48:07.040
And so this inequality there
would cap the number of steps
00:48:07.040 --> 00:48:08.570
you would have to do.
00:48:08.570 --> 00:48:11.660
You would never actually end
up having too many iterations.
00:48:16.640 --> 00:48:19.730
So this is a discussion of
the sparse regularity lemma.
00:48:19.730 --> 00:48:21.980
And the main message here
is that the regularity lemma
00:48:21.980 --> 00:48:23.990
itself is not so difficult--
00:48:23.990 --> 00:48:27.090
that's largely the same as
Szemeredi's regularity lemma.
00:48:27.090 --> 00:48:29.270
And so that's actually not
the most difficult part
00:48:29.270 --> 00:48:31.580
of sparse triangle
removal lemma.
00:48:31.580 --> 00:48:35.840
The difficulty lies in the other
step in the regularity method--
00:48:35.840 --> 00:48:38.220
namely, the counting step.
00:48:38.220 --> 00:48:40.820
And we already alluded
to this in the past.
00:48:40.820 --> 00:48:43.010
The point is that there
is no counting lemma
00:48:43.010 --> 00:48:46.540
for sparse regular graphs.
00:48:46.540 --> 00:48:49.150
And we already saw
an example where,
00:48:49.150 --> 00:48:52.750
if you start with a
random graph which
00:48:52.750 --> 00:48:55.030
has a small number
of triangles and I
00:48:55.030 --> 00:48:57.760
delete a small number
of edges corresponding
00:48:57.760 --> 00:48:59.590
to those triangles--
00:48:59.590 --> 00:49:04.300
one, I do not affect
its quasi-randomness.
00:49:04.300 --> 00:49:06.040
But two, there's no
triangles anymore,
00:49:06.040 --> 00:49:08.660
so there's no triangle
counting lemma.
00:49:08.660 --> 00:49:11.290
And that's a serious obstacle,
because you need this counting
00:49:11.290 --> 00:49:12.560
step.
00:49:12.560 --> 00:49:15.100
So what I would
like to explain next
00:49:15.100 --> 00:49:20.190
is how you can salvage that
and use this hypothesis here
00:49:20.190 --> 00:49:24.040
written in yellow to obtain a
counting lemma so that you can
00:49:24.040 --> 00:49:26.290
complete this
regularity method that
00:49:26.290 --> 00:49:29.200
would allow you to prove the
sparse triangle removal lemma.
00:49:29.200 --> 00:49:31.210
And a similar kind of
technique can allow you
00:49:31.210 --> 00:49:33.860
to do the Green-Tao theorem.
00:49:33.860 --> 00:49:36.970
So let's take a quick break.
00:49:36.970 --> 00:49:38.140
OK, any questions so far.
00:49:41.695 --> 00:49:43.320
So let's talk about
the counting lemma.
00:49:51.940 --> 00:49:55.240
So, the first case of the
counting lemma we considered
00:49:55.240 --> 00:49:56.710
was the triangle counting lemma.
00:50:03.170 --> 00:50:05.190
So remember what it says.
00:50:05.190 --> 00:50:08.870
If you have 3 vertex sets--
00:50:08.870 --> 00:50:18.090
V1, V2, V3-- such that,
between each pair,
00:50:18.090 --> 00:50:19.770
it is epsilon regular.
00:50:24.880 --> 00:50:32.100
And edge density--
that's for simplicity's
00:50:32.100 --> 00:50:35.090
sake-- they all have
the same edge density.
00:50:35.090 --> 00:50:36.690
Actually, they can be different.
00:50:36.690 --> 00:50:40.530
So d sub ij-- so possibly
different edge densities.
00:50:40.530 --> 00:50:42.120
But I have the set-up.
00:50:42.120 --> 00:50:44.610
And then the triangle
counting lemma
00:50:44.610 --> 00:50:51.220
tells us that the
number of triangles
00:50:51.220 --> 00:50:59.280
with one vertex in
each part is basically
00:50:59.280 --> 00:51:02.730
what you would expect
in the random case--
00:51:02.730 --> 00:51:05.110
namely, multiplying
these three edge
00:51:05.110 --> 00:51:10.530
densities together,
plus a small error,
00:51:10.530 --> 00:51:15.880
and then multiplying the
vertex sets' sizes together.
00:51:20.050 --> 00:51:22.450
So what we would
like is a statement
00:51:22.450 --> 00:51:28.610
that says that if you
have epsilon p regular
00:51:28.610 --> 00:51:33.590
and x densities now
at scale p, then
00:51:33.590 --> 00:51:36.610
we would want the
same thing to be true.
00:51:36.610 --> 00:51:39.140
Here, I should add
an extra p cubed,
00:51:39.140 --> 00:51:42.040
because that's the densities
we're working with.
00:51:42.040 --> 00:51:44.910
And I want some error here--
00:51:44.910 --> 00:51:48.200
OK, I can even let you
take some other epsilon.
00:51:48.200 --> 00:51:52.550
But small changes are OK.
00:51:52.550 --> 00:51:54.610
So that's the kind of
statement we want--
00:51:54.610 --> 00:51:56.250
and this is false.
00:51:56.250 --> 00:51:58.490
So this is completely false.
00:51:58.490 --> 00:52:00.740
And the example
that I said earlier
00:52:00.740 --> 00:52:07.130
was one of these examples
where you have a random graph.
00:52:07.130 --> 00:52:11.810
So this initial
version is false,
00:52:11.810 --> 00:52:18.870
because if you take a G and
p, with p somewhat less than 1
00:52:18.870 --> 00:52:24.080
over root n, and then remove
an edge from each triangle--
00:52:24.080 --> 00:52:26.010
or just remove all
the triangles--
00:52:26.010 --> 00:52:30.920
then you have a graph which
is still fairly pseudo-random,
00:52:30.920 --> 00:52:33.600
but it has no triangles.
00:52:33.600 --> 00:52:36.170
So you cannot have
a counting lemma.
00:52:36.170 --> 00:52:38.300
So there's another example
which, in some sense,
00:52:38.300 --> 00:52:40.590
is even better than
this random example.
00:52:40.590 --> 00:52:42.690
And it's a somewhat
mysterious example
00:52:42.690 --> 00:52:51.530
due to a law that gives
you a pseudo-random gamma.
00:52:51.530 --> 00:52:55.070
So it's, in some sense, an
optimally pseudo-random gamma,
00:52:55.070 --> 00:53:07.100
such that it is d-regular with
d on the order of n to the 3/2s.
00:53:07.100 --> 00:53:11.870
And it's an nd lambda
graph, where lambda
00:53:11.870 --> 00:53:15.340
is on the order of root d.
00:53:15.340 --> 00:53:16.900
Because here, d
is not a constant.
00:53:16.900 --> 00:53:19.060
But even in this case,
roughly speaking,
00:53:19.060 --> 00:53:23.030
this is as pseudo-random
as you can expect.
00:53:23.030 --> 00:53:26.600
So the second eigenvalue
is roughly square root
00:53:26.600 --> 00:53:28.610
of the degree.
00:53:28.610 --> 00:53:30.830
And yet, this graph
is triangle free.
00:53:37.420 --> 00:53:40.510
So you have some graph which,
for all the other kinds
00:53:40.510 --> 00:53:42.910
of pseudo-randomness
is very nice.
00:53:42.910 --> 00:53:45.650
So it has all the nice
pseudo-randomness properties,
00:53:45.650 --> 00:53:46.900
yet it is still triangle free.
00:53:46.900 --> 00:53:47.400
It's sparse.
00:53:50.190 --> 00:53:52.330
So the triangle
counting lemma is not
00:53:52.330 --> 00:53:54.040
true without
additional hypotheses.
00:53:56.660 --> 00:54:00.230
So I would like to add in some
hypotheses to make it true.
00:54:00.230 --> 00:54:03.010
And I would like a theorem.
00:54:03.010 --> 00:54:08.710
So again, I'm going to put
as a meta-theorem, which
00:54:08.710 --> 00:54:17.700
says that if you assume that G
is a subgraph of a sufficiently
00:54:17.700 --> 00:54:36.310
pseudo-random gamma and
gamma has edge density p,
00:54:36.310 --> 00:54:37.750
then the conclusion is true.
00:54:40.383 --> 00:54:41.550
And this is indeed the case.
00:54:47.640 --> 00:54:50.870
And I would like to tell
you what is the sufficiently
00:54:50.870 --> 00:54:52.310
pseudo-random--
00:54:52.310 --> 00:54:53.810
what does that hypothesis mean?
00:54:53.810 --> 00:54:56.600
So that at least you have
some complete theorem to take.
00:55:00.408 --> 00:55:02.200
There are several
versions of this theorem,
00:55:02.200 --> 00:55:05.110
so let me give you one
which I really like,
00:55:05.110 --> 00:55:08.910
because it has a fairly
clean hypothesis.
00:55:08.910 --> 00:55:14.350
And the version is that the
pseudo-randomness condition--
00:55:14.350 --> 00:55:16.840
so here it is.
00:55:16.840 --> 00:55:23.770
So, a sufficient
pseudo-randomness hypothesis
00:55:23.770 --> 00:55:35.550
on gamma, which is that gamma
has the correct number--
00:55:35.550 --> 00:55:39.730
"correct" in quotes, because
this is somewhat normative.
00:55:39.730 --> 00:55:42.040
So what I'm really
saying is it has,
00:55:42.040 --> 00:55:45.040
compared to a random case,
what you would expect.
00:55:45.040 --> 00:55:55.390
Densities of all
subgraphs of K--
00:55:55.390 --> 00:55:56.575
2, 2, 2.
00:56:15.160 --> 00:56:19.750
Having correct density of
H means having H density.
00:56:22.720 --> 00:56:30.130
1 plus little 1 times p, raised
to the number of edges of H,
00:56:30.130 --> 00:56:32.290
which is what you would
expect in a random case.
00:56:36.230 --> 00:56:38.540
So you should think
of there, again,
00:56:38.540 --> 00:56:41.265
not being just one graph,
but a sequence of graphs.
00:56:41.265 --> 00:56:42.890
You can also equivalently
write it down
00:56:42.890 --> 00:56:45.380
in terms of deltas and epsilons
having error parameters.
00:56:45.380 --> 00:56:47.900
But I like to think of it
having a sequence of graphs,
00:56:47.900 --> 00:56:51.260
just as in what we did
for quasi-random graphs.
00:56:51.260 --> 00:56:56.540
If your gamma has this
pseudo-randomness condition,
00:56:56.540 --> 00:57:00.688
which is we're in
this sparse setting.
00:57:00.688 --> 00:57:02.230
So if you try to
compare this to what
00:57:02.230 --> 00:57:05.260
we did for quasi-random
graphs, you might get confused.
00:57:05.260 --> 00:57:07.360
Because there, having
the correct C4 count
00:57:07.360 --> 00:57:09.520
already implies everything.
00:57:09.520 --> 00:57:11.710
This condition, it
actually does already
00:57:11.710 --> 00:57:15.464
include having the
correct C4 count.
00:57:15.464 --> 00:57:19.140
So K 2, 2, 2 is this
graph over here.
00:57:21.730 --> 00:57:27.480
And I'm saying that if it
has the correct density of H,
00:57:27.480 --> 00:57:38.450
whenever H is a
subgraph, of K 2, 2, 2--
00:57:38.450 --> 00:57:41.880
then it has a correct density.
00:57:41.880 --> 00:57:45.552
So in particular, it already
has a C4 count, but I want more.
00:57:45.552 --> 00:57:47.350
And it turns out this
is genuinely more,
00:57:47.350 --> 00:57:50.110
because in a sparse setting,
having the correct C4
00:57:50.110 --> 00:57:55.431
count is not equivalent to other
notions of pseudo-randomness.
00:57:58.380 --> 00:58:00.520
So this is a hypothesis.
00:58:00.520 --> 00:58:03.256
So if I start with a
sequence of gammas,
00:58:03.256 --> 00:58:07.190
I have the correct
counts of K 2, 2, 2s
00:58:07.190 --> 00:58:09.860
as well as subgraphs
of K 2, 2, 2s.
00:58:09.860 --> 00:58:14.210
Then I claim that that
pseudo-random host is good
00:58:14.210 --> 00:58:18.260
enough to have a
counting lemma--
00:58:18.260 --> 00:58:19.691
at least for triangles.
00:58:22.990 --> 00:58:24.067
Any questions?
00:58:30.150 --> 00:58:33.190
Now, you might want to ask for
some intuitions about where
00:58:33.190 --> 00:58:35.230
this condition comes from.
00:58:35.230 --> 00:58:38.460
The proof itself
takes a few pages.
00:58:38.460 --> 00:58:39.850
I won't try to do it here.
00:58:39.850 --> 00:58:44.080
I might try to give you some
intuition how the proof might
00:58:44.080 --> 00:58:46.090
go and also what
are the difficulties
00:58:46.090 --> 00:58:50.520
you might run into when you
try to execute this proof.
00:58:50.520 --> 00:58:54.000
But, at least how
I think of it is
00:58:54.000 --> 00:59:00.930
that this K 2, 2, 2 condition
plays a role similar to how
00:59:00.930 --> 00:59:03.870
previously, in dense
quasi-random graphs,
00:59:03.870 --> 00:59:06.540
we had this somewhat
magical looking
00:59:06.540 --> 00:59:09.860
C4 condition,
which can be viewed
00:59:09.860 --> 00:59:16.280
as a doubled version of an edge.
00:59:16.280 --> 00:59:19.430
So actually, the technical
name is called a blow-up.
00:59:19.430 --> 00:59:22.180
It's a blow-up of an edge.
00:59:22.180 --> 00:59:30.280
Whereas the K 2, 2, 2 condition
is a 2 blow-up of a triangle.
00:59:35.090 --> 00:59:41.210
And this 2 blow-up hypothesis is
some kind of a graph theoretic
00:59:41.210 --> 00:59:43.220
analogue of controlling
second moment.
00:59:53.720 --> 00:59:56.390
Just as knowing the variance
of a random variable--
00:59:56.390 --> 00:59:57.920
knowing its second moment--
00:59:57.920 --> 01:00:00.200
helps you to control
the concentration
01:00:00.200 --> 01:00:03.770
of that random variable, showing
that it's fairly concentrated.
01:00:03.770 --> 01:00:07.130
And it turns out that having
this graphical second moment
01:00:07.130 --> 01:00:11.600
in this sense also allows you
to control its properties so
01:00:11.600 --> 01:00:14.610
that you can have nice tools,
like the counting lemma.
01:00:20.500 --> 01:00:22.450
So let me explain some
of the difficulties.
01:00:22.450 --> 01:00:26.390
If you try to run the original
proof of the triangle removal
01:00:26.390 --> 01:00:31.010
lemma for the sparse
setting, what happens?
01:00:31.010 --> 01:00:36.690
So if you start with a vertex--
01:00:36.690 --> 01:00:39.940
so remember how the proof of
triangle removal lemma went.
01:00:39.940 --> 01:00:46.520
You start with this set-up
and you pick a typical vertex.
01:00:46.520 --> 01:00:51.390
This typical vertex has lots
of neighbors to the left
01:00:51.390 --> 01:00:53.980
and lots of neighbors
to the right.
01:00:53.980 --> 01:00:58.290
And here, a lot means
roughly the edge density
01:00:58.290 --> 01:01:00.570
times the number of vertices--
01:01:00.570 --> 01:01:04.630
and a lot of vertices over here.
01:01:04.630 --> 01:01:06.780
And then you say that,
because these are two fairly
01:01:06.780 --> 01:01:11.280
large vertex sets, there are
lots of edges between them
01:01:11.280 --> 01:01:16.980
by the hypotheses on
epsilon regularity,
01:01:16.980 --> 01:01:19.660
between the bottom two sets.
01:01:19.660 --> 01:01:22.120
But now, in the
sparse setting, we
01:01:22.120 --> 01:01:26.580
have an additional factor of p.
01:01:26.580 --> 01:01:31.180
So these two sets
are now quite small.
01:01:31.180 --> 01:01:33.040
They're much smaller
than what you
01:01:33.040 --> 01:01:36.280
can guarantee from the
definition of epsilon,
01:01:36.280 --> 01:01:38.170
p regular.
01:01:38.170 --> 01:01:42.340
So you cannot conclude from
them being epsilon regular that
01:01:42.340 --> 01:01:46.120
there are enough edges between
these two very small sets.
01:01:46.120 --> 01:01:49.390
So the strategy of proving
the triangle removal lemma
01:01:49.390 --> 01:01:51.410
breaks down in the
sparse setting.
01:02:00.950 --> 01:02:04.040
In general-- not
just for triangles,
01:02:04.040 --> 01:02:09.010
but for other H's as well--
01:02:09.010 --> 01:02:12.760
we also have this
counting lemma.
01:02:12.760 --> 01:02:14.210
So, the sparse counting lemma.
01:02:22.245 --> 01:02:24.990
And also the triangle case,
which I stated earlier.
01:02:24.990 --> 01:02:29.660
So this is drawing work due
to David Colin, Jacob Fox,
01:02:29.660 --> 01:02:31.820
and myself.
01:02:31.820 --> 01:02:33.840
Says that there
is a county lemma.
01:02:33.840 --> 01:02:35.370
So let me be very informal.
01:02:35.370 --> 01:02:52.230
So, that there exists a sparse
counting lemma for counting H,
01:02:52.230 --> 01:02:56.580
in this set-up as before.
01:02:56.580 --> 01:03:10.640
If gamma has a
pseudo-random property
01:03:10.640 --> 01:03:29.140
of containing the correct
density of all subgraphs
01:03:29.140 --> 01:03:40.260
of the 2 blow-up of H.
01:03:40.260 --> 01:03:44.370
Just as in the triangle,
the 2 blow-up is K 2, 2, 2.
01:03:44.370 --> 01:03:52.950
In general, the 2 blow-up
takes a graph, H, and then
01:03:52.950 --> 01:03:58.920
doubles every vertex
and puts in four edges
01:03:58.920 --> 01:04:02.940
between each pair of vertices.
01:04:02.940 --> 01:04:11.420
So that's the 2 blow-up of H.
01:04:11.420 --> 01:04:16.010
If your gamma has pseudo-random
properties concerning
01:04:16.010 --> 01:04:18.530
counting subgraphs
of this 2 blow-up,
01:04:18.530 --> 01:04:22.850
then you can obtain a
counting lemma for H itself.
01:04:26.710 --> 01:04:27.628
Any questions?
01:04:32.410 --> 01:04:35.090
OK, so let's take this counting
lemma for granted for now.
01:04:38.690 --> 01:04:45.760
How do we proceed to proving the
sparse triangle removal lemma?
01:04:45.760 --> 01:04:48.730
Well, I claim that actually
it's the same proof where
01:04:48.730 --> 01:04:52.840
you run the usual simulated
regularity proof of triangle
01:04:52.840 --> 01:04:53.980
removal lemma.
01:04:53.980 --> 01:04:55.840
But now, with all
of these extra tools
01:04:55.840 --> 01:04:58.180
and these extra
hypotheses, you then
01:04:58.180 --> 01:05:01.100
would obtain the
sparse triangle removal
01:05:01.100 --> 01:05:03.640
lemma, which I stated earlier.
01:05:03.640 --> 01:05:05.560
And the hypothesis
that I left out--
01:05:05.560 --> 01:05:08.540
the sufficiently pseudo-random
hypothesis on gamma--
01:05:08.540 --> 01:05:12.520
is precisely this
hypothesis over here,
01:05:12.520 --> 01:05:14.070
as required by the
counting lemma.
01:05:23.370 --> 01:05:25.560
And once you have
that, then you can
01:05:25.560 --> 01:05:31.980
proceed to prove a relative
version of Roth's theorem--
01:05:31.980 --> 01:05:35.070
and also, by extension,
two hyper-graphs--
01:05:35.070 --> 01:05:37.953
also a relative version
of Szemeredi's theorem.
01:05:41.200 --> 01:05:43.680
So, recall that the Roth's
theorem tells you that if you
01:05:43.680 --> 01:05:45.540
have a sufficiently large--
01:05:45.540 --> 01:05:49.920
so let me first write
down Roth's theorem.
01:05:49.920 --> 01:05:53.440
And then I'll add in the extra
relative things in yellow.
01:05:53.440 --> 01:05:59.340
So if I start with A,
the subset of z mod N,
01:05:59.340 --> 01:06:06.640
such that A has size--
01:06:06.640 --> 01:06:08.170
at least delta n.
01:06:14.500 --> 01:06:16.870
So then, Roth's
theorem tells us that A
01:06:16.870 --> 01:06:19.780
contains at least one three-term
arithmetic progression.
01:06:19.780 --> 01:06:23.020
But actually, you can
boost that theorem.
01:06:23.020 --> 01:06:25.720
And you've seen some
examples of this in homework.
01:06:25.720 --> 01:06:27.910
And also our proofs also
do this exact same thing.
01:06:27.910 --> 01:06:30.640
If you look at any of the
proofs that we've seen so far,
01:06:30.640 --> 01:06:35.110
it tells us that A not only
contains one single 3Ap,
01:06:35.110 --> 01:06:46.070
but it contains many
3Ap's, where C is
01:06:46.070 --> 01:06:48.693
some number that is positive.
01:06:53.700 --> 01:06:55.470
So you can obtain
this by the versions
01:06:55.470 --> 01:06:58.273
we've seen before, either
by looking at a proof--
01:06:58.273 --> 01:06:59.940
problem is in the the
proof gift stack--
01:06:59.940 --> 01:07:04.275
or by using the black box
version of Roth's theorem.
01:07:04.275 --> 01:07:06.150
And then there's a super
saturation argument,
01:07:06.150 --> 01:07:10.220
which is similar to things
you've done in the homework.
01:07:10.220 --> 01:07:12.390
What we would like is
a relative version.
01:07:16.130 --> 01:07:22.020
And a relative version will
say that if you have a set, S,
01:07:22.020 --> 01:07:24.500
which is sufficiently
pseudo-random.
01:07:32.000 --> 01:07:35.390
And S has density, p.
01:07:38.990 --> 01:07:40.600
Here, [INAUDIBLE].
01:07:43.860 --> 01:07:54.260
And now A is a subset of S.
And A has size at least delta,
01:07:54.260 --> 01:08:01.280
that of S. Then, A contains
still lots of 3Ap's, but I
01:08:01.280 --> 01:08:03.580
need to modify the
quantity, because I
01:08:03.580 --> 01:08:04.880
am looking at density, p.
01:08:09.580 --> 01:08:11.920
So this statement is
also true if you're
01:08:11.920 --> 01:08:15.470
putting the appropriate
hypothesis into sufficiently
01:08:15.470 --> 01:08:17.560
pseudo-random.
01:08:17.560 --> 01:08:21.149
And what should
those hypotheses be?
01:08:21.149 --> 01:08:23.670
So think about the proof
of Roth's theorem--
01:08:23.670 --> 01:08:25.770
the one that we've done--
01:08:25.770 --> 01:08:27.120
where you set up a graph.
01:08:30.620 --> 01:08:31.710
So, you set up this graph.
01:08:34.760 --> 01:08:38.779
So, one way to do
this is that you
01:08:38.779 --> 01:08:42.920
say that you put in edges
between the three parts--
01:08:42.920 --> 01:08:44.930
x, y, and z.
01:08:44.930 --> 01:08:49.830
So the vertex sets are
all given by z mod N.
01:08:49.830 --> 01:09:03.770
And you put in an edge between
x and y, if 2x plus y lies in S.
01:09:03.770 --> 01:09:07.819
Pulling the edge between
x and z-- if x minus z
01:09:07.819 --> 01:09:11.420
lies in S and a third
edge between y and z,
01:09:11.420 --> 01:09:19.420
if minus y minus 2z
lies in S. So this
01:09:19.420 --> 01:09:24.220
is a graph that we constructed
in the proof of Roth's theorem.
01:09:24.220 --> 01:09:29.520
And when you construct this
graph, either for S or for A--
01:09:29.520 --> 01:09:30.760
as we did before--
01:09:30.760 --> 01:09:34.710
then we see that the
triangles in this graph
01:09:34.710 --> 01:09:38.720
correspond precisely to
the 3Ap's in the set.
01:09:41.550 --> 01:09:46.380
So, looking at the triangle
counting lemma and triangle
01:09:46.380 --> 01:09:48.330
removal lemma-- the
sparse versions--
01:09:48.330 --> 01:09:52.290
then you can read out what type
of pseudo-randomness conditions
01:09:52.290 --> 01:09:55.000
you would like on S--
01:09:55.000 --> 01:09:57.690
so, from this graph.
01:09:57.690 --> 01:10:00.330
So, we would like
a condition, which
01:10:00.330 --> 01:10:02.400
says that this graph here--
01:10:08.550 --> 01:10:11.570
which we'll call gamma sub S--
01:10:11.570 --> 01:10:21.123
to have the earlier
pseudo-randomness hypotheses.
01:10:26.060 --> 01:10:29.278
And you can spell this out.
01:10:29.278 --> 01:10:30.390
And let's do that.
01:10:30.390 --> 01:10:31.640
Let's actually spell this out.
01:10:31.640 --> 01:10:33.940
So what does this mean?
01:10:33.940 --> 01:10:38.610
What I mean is S, being
a subset of Z mod N--
01:10:38.610 --> 01:10:45.860
we say that it satisfies
what's called a 3-linear forms
01:10:45.860 --> 01:10:46.862
condition.
01:10:55.226 --> 01:11:03.770
If, for uniformly
chosen random x0,
01:11:03.770 --> 01:11:11.637
x1, y0, y1, z0, z1
elements of z mod nz.
01:11:19.440 --> 01:11:21.540
Think about this K 2, 2, 2.
01:11:21.540 --> 01:11:23.680
So draw a K 2, 2, 2 up there.
01:11:23.680 --> 01:11:26.330
So what are the edges
corresponding to the K 2, 2, 2?
01:11:26.330 --> 01:11:29.940
So they correspond to the
following expressions--
01:11:29.940 --> 01:11:33.740
minus y0 minus 2z0--
01:11:33.740 --> 01:11:37.170
minus y1 minus 2z0--
01:11:37.170 --> 01:11:40.560
minus y0 minus 2z1--
01:11:40.560 --> 01:11:43.070
minus y1 minus 2z1.
01:11:43.070 --> 01:11:47.170
So those are the edges
corresponding to the bottom.
01:11:47.170 --> 01:11:51.680
Draw C4 across the
bottom two vertex sets.
01:11:51.680 --> 01:11:54.510
But then there are
two more columns.
01:11:54.510 --> 01:11:56.370
And I'll just write
some examples,
01:11:56.370 --> 01:11:57.690
but you can fill in the rest.
01:12:06.670 --> 01:12:11.480
OK, so there are at
least 12 expressions.
01:12:11.480 --> 01:12:24.460
And what we would like is that,
for random, the probability
01:12:24.460 --> 01:12:30.340
that all of these numbers
are contained in S
01:12:30.340 --> 01:12:44.434
is within 1 plus little 1
factor of the expectation,
01:12:44.434 --> 01:12:48.220
if S were a random set.
01:12:54.700 --> 01:12:58.130
In other words, in this
case, it's p raised to 12--
01:12:58.130 --> 01:13:04.980
random set of density, p.
01:13:04.980 --> 01:13:22.110
And furthermore, the same
holds if any subset of these 12
01:13:22.110 --> 01:13:23.550
expressions are erased.
01:13:40.270 --> 01:13:42.740
Now, I want you to use your
imagination and think about
01:13:42.740 --> 01:13:48.090
what the theorem would look like
for not 3Ap's, but for 4Ap's--
01:13:48.090 --> 01:13:49.960
and also for k-Ap's in general.
01:13:49.960 --> 01:14:03.350
So there is a relative Szemeredi
theorem, which tells you
01:14:03.350 --> 01:14:08.080
that if you start with S--
01:14:08.080 --> 01:14:15.460
so here, we fix K. If
you start with this S,
01:14:15.460 --> 01:14:19.177
that satisfies the
k-linear forms condition.
01:14:24.050 --> 01:14:31.840
And A is a subset of S
that is fairly large.
01:14:36.760 --> 01:14:43.350
Then A has k-Ap.
01:14:43.350 --> 01:14:45.310
So I'm being
slightly sloppy here,
01:14:45.310 --> 01:14:47.050
but that's the spirit
of the theorem--
01:14:47.050 --> 01:14:48.880
that you have this
Szemeredi theorem
01:14:48.880 --> 01:14:51.050
inside a sparse
pseudo-random set,
01:14:51.050 --> 01:14:53.520
as long as the
pseudo-random set satisfies
01:14:53.520 --> 01:14:55.290
this k-linear forms condition.
01:14:55.290 --> 01:14:57.110
And that k-linear
forms condition
01:14:57.110 --> 01:14:59.740
is an extension of this
3-linear forms condition, where
01:14:59.740 --> 01:15:04.120
you write down the proof that
we saw for Szemeredi's theorem,
01:15:04.120 --> 01:15:05.560
using hyper-graphs.
01:15:05.560 --> 01:15:07.660
Write down the
corresponding linear forms--
01:15:07.660 --> 01:15:10.736
you expand them out and then
you write down this statement.
01:15:18.090 --> 01:15:23.180
So this is basically what
I did for my PhD thesis.
01:15:23.180 --> 01:15:25.510
So we can ask, well, what
did Green and Tao do?
01:15:25.510 --> 01:15:28.990
So they had the original
theorem back in 2006.
01:15:28.990 --> 01:15:32.350
So their theorem, which also was
a relative Szemeredi theorem,
01:15:32.350 --> 01:15:35.380
has some additional,
more technical hypotheses
01:15:35.380 --> 01:15:39.190
known as correlation conditions,
which I won't get into.
01:15:39.190 --> 01:15:43.990
But at the end of the day, they
constructed these pseudoprimes.
01:15:43.990 --> 01:15:48.280
And then they verified that
those pseudoprimes satisfied
01:15:48.280 --> 01:15:53.410
these required
pseudo-randomness hypotheses--
01:15:53.410 --> 01:15:56.660
that those pseudoprimes
satisfied these linear forms
01:15:56.660 --> 01:16:00.590
conditions, as well as
their now-extraneous
01:16:00.590 --> 01:16:03.630
additional pseudo-randomness
hypotheses.
01:16:03.630 --> 01:16:06.530
And then combining this
combinatorial theorem
01:16:06.530 --> 01:16:08.210
with that number
adiabatic result.
01:16:08.210 --> 01:16:14.760
You put them together, you
obtain the Green-Tao theorem,
01:16:14.760 --> 01:16:18.860
which tells you not just that
the primes contain arbitrarily
01:16:18.860 --> 01:16:22.910
long arithmetic progressions,
but any positive density
01:16:22.910 --> 01:16:26.690
subset of the primes
also contains arbitrarily
01:16:26.690 --> 01:16:30.245
long arithmetic progressions.
01:16:30.245 --> 01:16:32.120
All of these theorems--
now, if you pass down
01:16:32.120 --> 01:16:34.900
to a relatively dense subset,
it still remains true.
01:16:38.180 --> 01:16:39.010
Any questions?
01:16:42.370 --> 01:16:43.850
So this is the general method.
01:16:43.850 --> 01:16:46.210
So the general method is you
have the sparse regularity
01:16:46.210 --> 01:16:46.900
method.
01:16:46.900 --> 01:16:49.540
And provided that you have
a good counting lemma,
01:16:49.540 --> 01:16:52.840
you can transfer the entire
method to the sparse setting.
01:16:52.840 --> 01:16:56.720
But getting the counting lemma
is often quite difficult.
01:16:56.720 --> 01:16:59.950
And there are still
interesting open problems--
01:16:59.950 --> 01:17:03.130
in particular, what kind of
pseudo-randomness hypotheses
01:17:03.130 --> 01:17:06.000
do we really need?
01:17:06.000 --> 01:17:08.430
Another thing is that
you don't actually
01:17:08.430 --> 01:17:10.960
have to go through
regularity yourself.
01:17:10.960 --> 01:17:12.930
So there is an additional
method-- which,
01:17:12.930 --> 01:17:16.710
unfortunately I don't
have time to discuss--
01:17:16.710 --> 01:17:23.040
called transference,
where the story I've told
01:17:23.040 --> 01:17:26.700
you is that you
look at the proof
01:17:26.700 --> 01:17:29.910
of Roth's theorem, the proof
of Szemeredi's theorem.
01:17:29.910 --> 01:17:32.850
And you transfer the
methods of those proofs
01:17:32.850 --> 01:17:34.740
to the sparse setting.
01:17:34.740 --> 01:17:36.260
And you can do that.
01:17:36.260 --> 01:17:39.200
But it turns out, you can
do something even better--
01:17:39.200 --> 01:17:42.050
is that you can
transfer the results.
01:17:42.050 --> 01:17:44.840
And this is what
happens in Green-Tao.
01:17:44.840 --> 01:17:49.520
If you look at Szemeredi's
theorem as a black-box theorem
01:17:49.520 --> 01:17:52.010
and you're happy
with its statement,
01:17:52.010 --> 01:17:56.570
you can use these
methods to transfer
01:17:56.570 --> 01:17:59.750
that result as a black
box without knowing
01:17:59.750 --> 01:18:04.520
its proof to the sparse
pseudo-random setting.
01:18:04.520 --> 01:18:07.520
And that sounds almost
too good to be true,
01:18:07.520 --> 01:18:10.310
but it's worth
seeing how it goes.
01:18:10.310 --> 01:18:14.520
And if you want to learn
more about this subject,
01:18:14.520 --> 01:18:22.560
there's a survey by Colin Fox
and by myself called "Green-Tao
01:18:22.560 --> 01:18:24.073
theorem, an exposition."
01:18:34.430 --> 01:18:38.980
So, where you'll find a
self-contained complete proof
01:18:38.980 --> 01:18:41.935
of the Green-Tao theorem,
except no modulo--
01:18:41.935 --> 01:18:43.560
the proof of Szemeredi's
theorem, which
01:18:43.560 --> 01:18:45.650
we've called as a black box.
01:18:45.650 --> 01:18:48.070
But you'll see how the
transference method
01:18:48.070 --> 01:18:48.843
works there.
01:18:48.843 --> 01:18:50.260
And it involves
many of the things
01:18:50.260 --> 01:18:52.580
that we've discussed
so far in this course,
01:18:52.580 --> 01:18:55.960
including discussions of the
regularity method, the counting
01:18:55.960 --> 01:18:56.470
lemma.
01:18:56.470 --> 01:19:00.580
And it will contain a proof of
this sparse triangle counting
01:19:00.580 --> 01:19:02.584
lemma.
01:19:02.584 --> 01:19:03.510
OK, good.
01:19:03.510 --> 01:19:05.390
We stop here.