Vapnik–Chervonenkis Density on Indiscernible Sequences, Stability, and the Maximum Property

Abstract

This paper presents some finite combinatorics of set systems with applications to model theory, particularly the study of dependent theories. There are two main results. First, we give a way of producing lower bounds on ${\mathrm{VC}}_{\mathrm{ind}}$-density and use it to compute the exact ${\mathrm{VC}}_{\mathrm{ind}}$-density of polynomial inequalities and a variety of geometric set families. The main technical tool used is the notion of a maximum set system, which we juxtapose to indiscernibles. In the second part of the paper we give a maximum set system analogue to Shelah’s characterization of stability using indiscernible sequences.