Combinatorial Theory: Introduction to Graph Theory, Extremal and Enumerative Combinatorics

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)