"There is no problem in all mathematics that cannot be solved by direct counting." - Ernst Mach
"I hope we'll be able to solve these problems before we leave." - Paul Erdos
April 1: Introduction, Bridges of Konigsberg
April 3: Characterization of Eulerian graphs, a note on Hamiltonicity
April 6: Basic properties of trees, definition of planarity
April 8: Euler formula, K5 is not planar
April 10: Chromatic number, a 6-color theorem for planar graphs
April 13: Subgraphs, induced subgraphs, minors
April 15: Clique number, independence number, Hadwiger's conjecture
April 17: Proof of the Four Color Theorem
April 20: Finishing fake proof of Four Color Theorem, a real proof of a Five Color Theorem,
more planarity theorems and Conway's thrackle conjecture
April 22: Introduction to Ramsey theory, definition of R(m,n) and some basic properties
April 24: Multicolor Ramsey numbers
April 27: Infinite multicolor hypergraph Ramsey theorem
April 29: Finite multicolor hypergraph Ramsey theorem, Happy Ending Theorem
May 1: Turan's theorem
May 4: Introduction to enumerative combinatorics, binomial coefficients
May 6: Binomial theorem and few of its consequences
May 8: (Exam)
May 11: Multinomial coefficients, unordered selections i.e. "stars and bars"
May 13: Fibonacci numbers, solving linear recurrances
May 15: Catalan numbers, and a nonlinear recurrance
May 18: Generating functions
May 20: Hook length formula, and applications
May 22: Cauchy-Frobenius-Polya-Burnside, counting under symmetry
May 27: Introduction to the probabilistic method: union bounds and expectation
May 29: Graphs with high girth and chromatic number
June 1: Crossing number lemma and the Szemeredi-Trotter theorem
(see also: Terry Tao's comments on his blog)
June 3: Three quickies: applications of the probabilistic method to graph theory, linear algebra, and additive number theory
Here is your take-home final. It is due by 5pm, on Friday June 5.