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. (shrink)