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

2 Additive Number Theory

Theorems of Schur and Van der Waerden

3 Lower Bound in Schur’s Theorem

Erdös-Szekeres Theorem (Two Proofs)

2-Colorability of Multigraphs

Intersection Conditions

4 More on Colorings

Greedy Algorithm

Height Functions Argument for 3-Colorings of a Rectangle

Erdös Theorem

5 More on Colorings (cont.)

Erdös-Lovász Theorem

Brooks Theorem

6 5-Color Theorem

Vizing’s Theorem

7 Edge Coloring of Bipartite Graphs

Heawood Formula

8 Glauber Dynamics

The Diameter

Explicit Calculations

Bounds on Chromatic Number via the Number of Edges, and via the Independence Number

9 Chromatic Polynomial

NBC Theorem

10 Acyclic Orientations

Stanley’s Theorem

Two Definitions of the Tutte Polynomial

11 More on Tutte Polynomial

Special Values

External and Internal Activities

Tutte’s Theorem

12 Tutte Polynomial for a Cycle

Gessel’s Formula for Tutte Polynomial of a Complete Graph

13 Crapo’s Bijection

Medial Graph and Two Type of Cuts

Introduction to Knot Theory

Reidemeister Moves

14 Kauffman Bracket and Jones Polynomial (PDF)
15 Linear Algebra Methods

Oddtown Theorem

Fisher’s Inequality

2-Distance Sets

16 Non-uniform Ray-Chaudhuri-Wilson Theorem

Frankl-Wilson Theorem

17 Borsuk Conjecture

Kahn-Kalai Theorem

18 Packing with Bipartite Graphs

Testing Matrix Multiplication

19 Hamiltonicity, Basic Results

Tutte’s Counter Example

Length of the Longest Path in a Planar Graph

20 Grinberg’s Formula

Lovász and Babai Conjectures for Vertex-transitive Graphs

Dirac’s Theorem

21 Tutte’s Theorem

Every Cubic Graph Contains either no HC, or At Least Three

Examples of Hamiltonian Cycles in Cayley Graphs of Sn

22 Hamiltonian Cayley Graphs of General Groups (PDF)
23 Menger Theorem

Gallai-Milgram Theorem

24 Dilworth Theorem

Hall’s Marriage Theorem

Erdös-Szekeres Theorem

25 Sperner Theorem

Two Proofs of Mantel Theorem

Graham-Kleitman Theorem

26 Swell Colorings

Ward-Szabo Theorem

Affine Planes

27 Turán’s Theorem

Asymptotic Analogues

28 Pattern Avoidance

The case of S3 and Catalan Numbers

Stanley-Wilf Conjecture

29 Permutation Patterns

Arratia Theorem

Furedi-Hajnal Conjecture

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

33 Solid Partitions

MacMahon’s Theorem

Hook-content Formula

34 Hook Length Formula (PDF)
35 Two Polytope Theorem (PDF)
36 Connection to RSK

Special Cases

37 Duality

Number of Involutions in Sn

38 Direct Bijective Proof of the Hook Length Formula (PDF)
39 Introduction to Tilings

Thurston’s Theorem


Course Info

As Taught In
Spring 2005