On the shatter function of semilinear set systems

Abstract

We show that the shatter function of a semilinear set system on ${\mathbb{R}}^m$ is asymptotic to a polynomial. This confirms, for the structure $(\mathbb{R};+,<)$, a conjecture of Chernikov and is a step towards characterizing model-theoretic linearity via shatter functions.

40 pages; minor edits