Vapnik-Chervonenkis density on indiscernible sequences, stability, and the maximum property

TL;DR

This paper explores VC_ind-density in set systems, providing lower bounds and exact calculations for polynomial inequalities and geometric families, and introduces a maximum set system analogue to Shelah's stability characterization using indiscernibles.

Contribution

It introduces a method to bound VC_ind-density, computes exact densities for specific set families, and develops a maximum set system analogue to stability characterization.

Findings

Exact VC_ind-density for polynomial inequalities

Lower bounds on VC_ind-density for geometric set families

Maximum set system analogue to stability

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_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.