VC-saturated set systems

TL;DR

This paper investigates the size of the smallest maximal set family with a given VC-dimension, providing bounds that are independent of the size of the ground set, through both random and explicit constructions.

Contribution

It establishes new bounds on the saturation number for VC-dimension $d$, showing it is at most $4^{d+1}$ regardless of the size of the universe.

Findings

Saturation number is at most $4^{d+1}$ for VC-dimension $d$

Constructs both random and explicit examples

Bounds are independent of the size of the ground set

Abstract

The well-known Sauer lemma states that a family $\mathcal{F}\subseteq 2^{[n]}$ of VC-dimension at most $d$ has size at most $\sum_{i=0}^d\binom{n}{i}$. We obtain both random and explicit constructions to prove that the corresponding saturation number, i.e., the size of the smallest maximal family with VC-dimension $d\ge 2$, is at most $4^{d+1}$, and thus is independent of $n$.