VC dimension and a union theorem for set systems

TL;DR

This paper generalizes a VC dimension bound for set systems involving symmetric differences, providing new extremal set size results and answering open questions in combinatorics.

Contribution

It extends the VC dimension union theorem to a k-fold setting and connects the problem to classical extremal set theory, settling previous open questions.

Findings

Bound on set system size involving VC dimension and symmetric differences

Construction of large set families with bounded VC dimension for intersections and unions

Resolution of open questions from prior work by Dvir and Moran

Abstract

Fix positive integers $k$ and $d$. We show that, as $n\to\infty$, any set system $\mathcal{A} \subset 2^{[n]}$ for which the VC dimension of $\{ \triangle_{i=1}^k S_i \mid S_i \in \mathcal{A}\}$ is at most $d$ has size at most $(2^{d\bmod{k}}+o(1))\binom{n}{\lfloor d/k\rfloor}$. Here $\triangle$ denotes the symmetric difference operator. This is a $k$-fold generalisation of a result of Dvir and Moran, and it settles one of their questions. A key insight is that, by a compression method, the problem is equivalent to an extremal set theoretic problem on $k$-wise intersection or union that was originally due to Erd\H{o}s and Frankl. We also give an example of a family $\mathcal{A} \subset 2^{[n]}$ such that the VC dimension of $\mathcal{A}\cap \mathcal{A}$ and of $\mathcal{A}\cup \mathcal{A}$ are both at most $d$, while $\lvert \mathcal{A} \rvert = \Omega(n^d)$. This provides a negative answer to another question of Dvir and Moran.