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

Calendar

Lec #	Topics	KEY DATES
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	Problem set 1 due
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	Problem set 2 due
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	Problem set 3 due
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	Problem set 4 due
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
23	Menger Theorem Gallai-Milgram Theorem	Problem set 5 due
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	Problem set 6 due
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	Problem set 7 due
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
35	Two Polytope Theorem
36	Connection to RSK Special Cases	Problem set 8 due
37	Duality Number of Involutions in Sn
38	Direct Bijective Proof of the Hook Length Formula
39	Introduction to Tilings Thurston’s Theorem