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 -density and use it to compute the exact -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.
"Vapnik–Chervonenkis Density on Indiscernible Sequences, Stability, and the Maximum Property." Notre Dame J. Formal Logic 56 (4) 583 - 593, 2015. https://doi.org/10.1215/00294527-3153597