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		 (This resource may not render correctly in a screen reader.PDF)
2 Additive Number Theory

Theorems of Schur and Van der Waerden		 (This resource may not render correctly in a screen reader.PDF)
3 Lower Bound in Schur's Theorem

Erdös-Szekeres Theorem (Two Proofs)

2-Colorability of Multigraphs

Intersection Conditions		 (This resource may not render correctly in a screen reader.PDF)
4 More on Colorings

Greedy Algorithm

Height Functions Argument for 3-Colorings of a Rectangle

Erdös Theorem		 (This resource may not render correctly in a screen reader.PDF)
5 More on Colorings (cont.)

Erdös-Lovász Theorem

Brooks Theorem		 (This resource may not render correctly in a screen reader.PDF)
6 5-Color Theorem

Vizing's Theorem		 (This resource may not render correctly in a screen reader.PDF)
7 Edge Coloring of Bipartite Graphs

Heawood Formula		 (This resource may not render correctly in a screen reader.PDF)
8 Glauber Dynamics

The Diameter

Explicit Calculations

Bounds on Chromatic Number via the Number of Edges, and via the Independence Number		 (This resource may not render correctly in a screen reader.PDF)
9 Chromatic Polynomial

NBC Theorem		 (This resource may not render correctly in a screen reader.PDF)
10 Acyclic Orientations

Stanley's Theorem

Two Definitions of the Tutte Polynomial		 (This resource may not render correctly in a screen reader.PDF)
11 More on Tutte Polynomial

Special Values

External and Internal Activities

Tutte's Theorem		 (This resource may not render correctly in a screen reader.PDF)
12 Tutte Polynomial for a Cycle

Gessel's Formula for Tutte Polynomial of a Complete Graph		 (This resource may not render correctly in a screen reader.PDF)
13 Crapo's Bijection

Medial Graph and Two Type of Cuts

Introduction to Knot Theory

Reidemeister Moves		 (This resource may not render correctly in a screen reader.PDF)
14 Kauffman Bracket and Jones Polynomial (This resource may not render correctly in a screen reader.PDF)
15 Linear Algebra Methods

Oddtown Theorem

Fisher's Inequality

2-Distance Sets		 (This resource may not render correctly in a screen reader.PDF)
16 Non-uniform Ray-Chaudhuri-Wilson Theorem

Frankl-Wilson Theorem		 (This resource may not render correctly in a screen reader.PDF)
17 Borsuk Conjecture

Kahn-Kalai Theorem		 (This resource may not render correctly in a screen reader.PDF)
18 Packing with Bipartite Graphs

Testing Matrix Multiplication		 (This resource may not render correctly in a screen reader.PDF)
19 Hamiltonicity, Basic Results

Tutte's Counter Example

Length of the Longest Path in a Planar Graph		 (This resource may not render correctly in a screen reader.PDF)
20 Grinberg's Formula

Lovász and Babai Conjectures for Vertex-transitive Graphs

Dirac's Theorem		 (This resource may not render correctly in a screen reader.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 		(This resource may not render correctly in a screen reader.PDF)
22 Hamiltonian Cayley Graphs of General Groups (This resource may not render correctly in a screen reader.PDF)
23 Menger Theorem

Gallai-Milgram Theorem		 (This resource may not render correctly in a screen reader.PDF)
24 Dilworth Theorem

Hall's Marriage Theorem

Erdös-Szekeres Theorem		 (This resource may not render correctly in a screen reader.PDF)
25 Sperner Theorem

Two Proofs of Mantel Theorem

Graham-Kleitman Theorem		 (This resource may not render correctly in a screen reader.PDF)
26 Swell Colorings

Ward-Szabo Theorem

Affine Planes		 (This resource may not render correctly in a screen reader.PDF)
27 Turán's Theorem

Asymptotic Analogues		 (This resource may not render correctly in a screen reader.PDF)
28 Pattern Avoidance

The case of S3 and Catalan Numbers

Stanley-Wilf Conjecture		 (This resource may not render correctly in a screen reader.PDF)
29 Permutation Patterns

Arratia Theorem

Furedi-Hajnal Conjecture		 (This resource may not render correctly in a screen reader.PDF)
30 Proof by Marcus and Tardos of the Stanley-Wilf Conjecture (This resource may not render correctly in a screen reader.PDF)
31 Non-intersecting Path Principle

Gessel-Viennot Determinants

Binet-Cauchy Identity		  
32 Convex Polyomino

Narayana Numbers

MacMahon Formula		 (This resource may not render correctly in a screen reader.PDF)
33 Solid Partitions

MacMahon's Theorem

Hook-content Formula		 (This resource may not render correctly in a screen reader.PDF)
34 Hook Length Formula (This resource may not render correctly in a screen reader.PDF)
35 Two Polytope Theorem (This resource may not render correctly in a screen reader.PDF)
36 Connection to RSK

Special Cases		 (This resource may not render correctly in a screen reader.PDF)
37 Duality

Number of Involutions in Sn 		(This resource may not render correctly in a screen reader.PDF)
38 Direct Bijective Proof of the Hook Length Formula (This resource may not render correctly in a screen reader.PDF)
39 Introduction to Tilings

Thurston's Theorem		 (This resource may not render correctly in a screen reader.PDF)