THE CLASSICS: Counting: The binomial theorem, selections with repetitions, partitions, double counting, the averaging principles. Advanced counting: Bounds on intersection size, Zarankiewicz's problem, Density of 0-1 matrices. Inclusion and Exclusion principle: The number of derangements. The pigeon-hole principle: The Erdos-Szekeres theorem, Mantel's theorem, Turan's theorem, Dirichlet's theorem.
EXTREMAL SET THEORY: Intersecting families: The sunflower lemma, The Erdos-Ko-Rado theorem. Chains and antichains: Dilworth's theorem, Sperner's theorem, Bollobas's theorem.
THE LINEAR ALGEBRA METHOD: A short introduction to some basic concepts of linear algebra. The basic method (using linear independence): Fisher's inequality, polynomial technique, Frankl-Wilson theorem, another proof of Bollobas's theorem. Orthogonality and rank arguments: Orthogonal coding, a bribery party, balanced families.
THE PROBABILISTIC METHOD: Basic tools. Counting sieve: Ramsey numbers, Van der Waerden's theorem, Tournaments, Kleitman-Spencer theorem. The Lovasz sieve: The local lemma and some of its applications.