The VC-dimension of random subsets of finite groups

TL;DR

This paper investigates the VC-dimension of random subsets in finite groups and establishes a law of large numbers for it as the subset size grows, addressing a question in combinatorial group theory.

Contribution

It provides the first law of large numbers for the VC-dimension of random subsets in finite groups, linking combinatorics and group theory.

Findings

Proves a law of large numbers for VC-dimension as N→∞

Answers a question posed by McDonald--Sahay--Wyman

Connects VC-dimension with properties of random Cayley graphs

Abstract

For a random subset of a finite group $G$ of cardinality $N$, we consider the VC-dimension of the family of its translates (equivalently the VC-dimension of a random Cayley graph) and prove a law of large numbers as $N\rightarrow\infty$. This answers a question of McDonald--Sahay--Wyman.