A Sauer-Shelah-Perles Lemma for Sumsets

TL;DR

This paper extends the Sauer-Shelah-Perles lemma to sumsets by establishing an upper bound on the size of set families based on VC dimension of their symmetric differences, using polynomial methods.

Contribution

It introduces a novel Sauer-Shelah-Perles type bound for sumsets involving VC dimension and demonstrates the limitations of similar bounds with union or intersection.

Findings

Bound on size of set families with VC dimension of symmetric differences

Polynomial method applied to combinatorial set theory

Limitations of analogous bounds for union and intersection

Abstract

We show that any family of subsets $A\subseteq 2^{[n]}$ satisfies $\lvert A\rvert \leq O\bigl(n^{\lceil{d}/{2}\rceil}\bigr)$, where $d$ is the VC dimension of $\{S\triangle T \,\vert\, S,T\in A\}$, and $\triangle$ is the symmetric difference operator. We also observe that replacing $\triangle$ by either $\cup$ or $\cap$ fails to satisfy an analogous statement. Our proof is based on the polynomial method; specifically, on an argument due to [Croot, Lev, Pach '17].