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)
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)

Course Info

Learning Resource Types

assignment Problem Sets