  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.
