# Blog Archives

# Topic Archive: NIP

# Ramsey-type theorems in certain NIP theories

In the paper “Crossing patterns of semi-algebraic sets” (J. Combin. Theory Ser. A 111, 2005) Alon et al. showed that families of graphs with the edge relation given by a semialgebraic relation of bounded complexity satisfy a stronger regularity property than arbitrary graphs. In this talk we show that this can be generalized to families of graphs whose edge relation is uniformly definable in a structure satisfying a certain model theoretic property called distality.

This is a joint work with A. Chernikov.

# VC-dimension in model theory and other subjects

Finite VC-dimension, a combinatorial property of families of

sets, was discovered simultaneously by Vapnik and Chervonenkis in the

context of probabilistic learning theory, and by Shelah in model

theory in the context of classification of unstable first-order

theories (where it is called NIP). From the model theoretic point of

view it is a very attractive setting generalizing stability and

o-minimality, and admitting a deep theory which had been recently used

to study ordered and valued fields. I will give an overview of some

results around NIP related to set theory (counting Dedekind cuts in

infinite linear orders), topological dynamics and compression schemes

in computational learning theory.

# On dense independent subsets of geometric structures

We consider expansions of geometric theories obtained by adding a predicate distinguishing a “dense” independent subset, generalizing a construction introduced by A. Dolich, C. Miller and C. Steinhorn in the o-minimal context. The expansion preserves many of the properties related to stability, simplicity, rosiness and NIP. We also study the structure induced on the predicate, and show that despite its geometric triviality, it inherits most of the “combinatorial” complexity of the original theory. This is a joint work with Alexander Berenstein.