These lecture notes were taken by Amanda Redlich, a student in the class, and were used with permission.
Lec # | Topics | Lecture Notes |
---|---|---|
1 |
Course Introduction
Ramsey Theorem |
(PDF) |
2 |
Additive Number Theory
Theorems of Schur and Van der Waerden |
(PDF) |
3 |
Lower Bound in Schur’s Theorem
Erdös-Szekeres Theorem (Two Proofs) 2-Colorability of Multigraphs Intersection Conditions |
(PDF) |
4 |
More on Colorings
Greedy Algorithm Height Functions Argument for 3-Colorings of a Rectangle Erdös Theorem |
(PDF) |
5 |
More on Colorings (cont.)
Erdös-Lovász Theorem Brooks Theorem |
(PDF) |
6 |
5-Color Theorem
Vizing’s Theorem |
(PDF) |
7 |
Edge Coloring of Bipartite Graphs
Heawood Formula |
(PDF) |
8 |
Glauber Dynamics
The Diameter Explicit Calculations Bounds on Chromatic Number via the Number of Edges, and via the Independence Number |
(PDF) |
9 |
Chromatic Polynomial
NBC Theorem |
(PDF) |
10 |
Acyclic Orientations
Stanley’s Theorem Two Definitions of the Tutte Polynomial |
(PDF) |
11 |
More on Tutte Polynomial
Special Values External and Internal Activities Tutte’s Theorem |
(PDF) |
12 |
Tutte Polynomial for a Cycle
Gessel’s Formula for Tutte Polynomial of a Complete Graph |
(PDF) |
13 |
Crapo’s Bijection
Medial Graph and Two Type of Cuts Introduction to Knot Theory Reidemeister Moves |
(PDF) |
14 | Kauffman Bracket and Jones Polynomial | (PDF) |
15 |
Linear Algebra Methods
Oddtown Theorem Fisher’s Inequality 2-Distance Sets |
(PDF) |
16 |
Non-uniform Ray-Chaudhuri-Wilson Theorem
Frankl-Wilson Theorem |
(PDF) |
17 |
Borsuk Conjecture
Kahn-Kalai Theorem |
(PDF) |
18 |
Packing with Bipartite Graphs
Testing Matrix Multiplication |
(PDF) |
19 |
Hamiltonicity, Basic Results
Tutte’s Counter Example Length of the Longest Path in a Planar Graph |
(PDF) |
20 |
Grinberg’s Formula
Lovász and Babai Conjectures for Vertex-transitive Graphs Dirac’s Theorem |
(PDF) |
21 |
Tutte’s Theorem
Every Cubic Graph Contains either no HC, or At Least Three Examples of Hamiltonian Cycles in Cayley Graphs of Sn |
(PDF) |
22 | Hamiltonian Cayley Graphs of General Groups | (PDF) |
23 |
Menger Theorem
Gallai-Milgram Theorem |
(PDF) |
24 |
Dilworth Theorem
Hall’s Marriage Theorem Erdös-Szekeres Theorem |
(PDF) |
25 |
Sperner Theorem
Two Proofs of Mantel Theorem Graham-Kleitman Theorem |
(PDF) |
26 |
Swell Colorings
Ward-Szabo Theorem Affine Planes |
(PDF) |
27 |
Turán’s Theorem
Asymptotic Analogues |
(PDF) |
28 |
Pattern Avoidance
The case of S3 and Catalan Numbers Stanley-Wilf Conjecture |
(PDF) |
29 |
Permutation Patterns
Arratia Theorem Furedi-Hajnal Conjecture |
(PDF) |
30 | Proof by Marcus and Tardos of the Stanley-Wilf Conjecture | (PDF) |
31 |
Non-intersecting Path Principle
Gessel-Viennot Determinants Binet-Cauchy Identity |
|
32 |
Convex Polyomino
Narayana Numbers MacMahon Formula |
(PDF) |
33 |
Solid Partitions
MacMahon’s Theorem Hook-content Formula |
(PDF) |
34 | Hook Length Formula | (PDF) |
35 | Two Polytope Theorem | (PDF) |
36 |
Connection to RSK
Special Cases |
(PDF) |
37 |
Duality
Number of Involutions in Sn |
(PDF) |
38 | Direct Bijective Proof of the Hook Length Formula | (PDF) |
39 |
Introduction to Tilings
Thurston’s Theorem |
(PDF) |