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 VC ind -density, and use it to compute the exact VC inddensity 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.1991 Mathematics Subject Classification. 03C45.

On Vapnik‐Chervonenkis density over indiscernible sequences

“…Quite recently, however, Johnson (cf. [, Corollary ]) has demonstrated a necessary and sufficient condition for the VC‐density and VC ind ‐density of a given formula to be equal.…”

“…These two values certainly do not coincide in general, for VC ind -density is always integer-valued, while VC-density can take noninteger values even in very simple scenarios. Quite recently, however, Johnson (see Corollary 3.5 of [7]) has demonstrated a necessary and sufficient condition for the VC-density and VC ind -density of a given formula to be equal.…”