Model theory and combinatorics
The following is a plan of a book in progress aiming to provide an itroduction to the recent interactions of model theory (centered around Shelah's classification and generalizations of stability) and combinatorics (additive, extremal, incidence and otherwise). Some chapters are partially written, all comments and suggestions are very welcome.- VC theory and NIP structures (Sauer-Shelah lemma, VC-theorem, (p,q)-theorem, VC-density calculations, examples of NIP structures, UDTFS and applications)
- Tame measure theory and hypergraph regularity (Szemeredi's regularity lemma and its strengthenings for NIP, stable, distal, Tao's algebraic regularity lemma and its variants)
- Pseudofinite dimension (Hrushovski-Wagner, Larsen-Pink, Erdos-Hajnal for stable graphs)
- Approximate subgroups and equivalence relations (Hrushovski, Massicot-Wagner, etc.)
- Incidence combinatorics in distal structures (Zarankiewicz, cutting lemmas, o-minimal and p-adic cases)
- Group configuration, Zilber's trichotomy and applications in combinatorics (expanding polynomials, Elekes-Szabo and generalizations, etc.)
- Large sets in NIP groups and additive Ramsey theory (B+C in NIP, etc.)