On the shatter function of semilinear set systems

TL;DR

This paper proves that the shatter function of semilinear set systems in real space grows asymptotically as a polynomial, confirming a conjecture for the structure of real numbers with addition and order.

Contribution

It establishes the polynomial growth of the shatter function for semilinear set systems, advancing the understanding of model-theoretic linearity.

Findings

Shatter function is asymptotic to a polynomial for semilinear set systems.

Confirms Chernikov's conjecture for the structure (; +, <).

Progress towards characterizing model-theoretic linearity.

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.