# Lecture Notes

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)

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)