VC-dimension and pseudo-random graphs

Abstract

Let $G$ be a graph and $U\subset V\left(G\right)$ be a set of vertices. For each $v\in U$, let $h_v:U\to \{0,1\}$ be the function defined by $h_v\left(u\right)=\left(\begin{array}{c}\&1\text{if}u\sim v,u\in U\\ 0\text{if}u⁄\sim v,u\in U,\end{array}\right)$and set $H\left(U\right)≔\{h_v:v\in U\}$. The first purpose of this paper is to study the following question: What families of graphs $G$ and what conditions on $U$ do we need so that the VC-dimension of $H\left(U\right)$ can be determined? We show that if $G$ is a pseudo-random graph, then under some mild conditions, the VC dimension of $H\left(U\right)$ can be bounded from below. Specific cases of this theorem recover and improve previous results on VC-dimension of functions defined by the well-studied distance and dot-product graphs over a finite field.