A uniform version of a theorem by Dvir and Moran

Abstract

Let $0\leq d\leq n$ be integers and $\mathcal{F}$ a family with ${\rm VCdim} ( \mathcal{F} \Delta \mathcal{F} )\leq d$. Analogously to the celebrated Sauer-Shelah Lemma, Dvir and Moran [6] proved that in this case $\mathcal{F}$ can have size at most $2\sum_{k=0}^{\lfloor d/2 \rfloor}\binom nk$. Our main result is a uniform version of this statement. We show that if $\mathcal{F}$ is a uniform family of subsets of [n] and ${\rm VCdim} ( \mathcal{F} \Delta \mathcal{F} )\leq d$, then

$$\left| \mathcal{F} \right|\le 2 {n \choose \lfloor d/2 \rfloor}.$$

Our proof is based on a uniform version of the Croot–Lev–Pach lemma, which we think can be of interest on its own.