Lecture notes: Here (updated April 21)
Using LaTeX: If you do not know how to use LaTeX, you can learn how using this excellent guide that was written by Kristin Heysse. The template that is referenced in the document is here and can be copy/pasted into a new Overleaf document (or a .tex file if you have TeX set up on your own machine).
Textbooks:
A Walk Through Combinatorics by Bona
Graph Theory by Bondy and Murty
Invitation to Discrete Mathematics by Matousek and Nesetril
Lecture notes on the probabilistic method by Matousek and Vondrak
Applied Combinatorics by Keller and Trotter
January 27: Introduction -- counting principles, first graph theory definitions and theorems
February 3: Graph theory basics -- trees, bipartite graphs.
February 10: planar graphs, graph coloring problems.
February 17: Introduction to extremal graph theory -- Mantel's and Turan's theorem, Ramsey's theorem
February 24: Erdos-Stone-Simonovits theorem, discussion of bipartite Turan numbers, Schur's theorem, Fermat's last theorem over finite fields. Introduction to Probabilistic method -- expectation and variance, first moment method, lower bounds for Turan and Ramsey numbers.
March 3: Markov and Chebyshev inequalities with applications
March 10: Chernoff bound, Lovasz Local Lemma, applications
March 17: Kovari-Sos-Turan theorem with number theory applications. Constructions of C_4-free graphs via combinatorial number theory and finite geometry
March 24: Sidon sets, algebraically defined graphs, orthogonal polarity graphs
March 31: Linear algebra preliminaries, the adjacency matrix, counting walks, structure from the eigenvalue equation, spectral Mantel's theorem
April 7: structure from the eigenvalue equation, spectral Mantel's theorem, Stanley's bound, Nosal's theorem, maximum spectral radius of F-free graphs, Moore graphs, Hoffman-Singleton theorem, strongly regular graphs
April 14: the expander mixing lemma, the Hoffman ratio bound, sum-product theorems in finite fields
April 21: the Graham-Pollak theorem, combinatorial nullstellensatz and applications
April 28: extremal set theory: oddtown, eventown, Erdos-Ko-Rado, 2-distance sets.
Homework 1 due February 5
Homework 2 due February 19
Homework 3 due February 26
Homework 4 due March 12
Homework 5 due March 19
Homework 6 due April 2
Homework 7 due April 9