Past lectures

Abstract

In many instances of combinatorial problems one studies special maps and correspondences which are tailored to capture a variety of discrete phenomena. Structural combinatorics studies such situations and stresses their relationship to the mainstream mathematics (model theory, algebra and probability in particular). This will be a self-contained course based on a forthcoming book.

The tentative contents

1. Mappings and Categories (Basic examples, concreteness as the only combinatorial obstacle -Freyd Vinarek Theorem)

2. Existence vrs. Counting (Reconstruction, cancellation in finite structures, Lovasz, Muller theorems, polynomials via maps)

3. Homomorphisms, tensions and flows (continuous and matroid setting, rich quasiorders)

4. Constructions (Product classes, dimension, LA method, product conjecture, applications, extension properties)

5. Coloring Order (density, independence, countable universality, bounds for minor closed families, countable graphs)

6. Homomorphism dualities (algorithmic and extremal aspects, Gallai-Roy type theorems)

7. Amalgamation and Combinatorial Sieve (density, sparse graphs, Ramsey theory)