Maximum shattering

TL;DR

This paper investigates the maximum number of size-d subsets of [n] that can be shattered by a family of size k, providing exact results for small d and asymptotic bounds for larger d, with applications to covering arrays.

Contribution

It develops a framework to analyze f(n,k,d), determines its exact value for d ≤ 2, and extends asymptotic results for larger d, improving understanding of shattering in combinatorics.

Findings

Exact value of f(n,k,d) for d ≤ 2.

Asymptotic formula for f(n,k,d) when d and n-d grow large.

Improved bounds for covering array existence.

Abstract

A family $\mathcal{F}$ of subsets of $[n]=\{1,2,\ldots,n\}$ shatters a set $A \subseteq [n]$ if for every $A' \subseteq A$ there is an $F \in \mathcal{F}$ such that $F \cap A=A'$. We develop a framework to analyze $f(n,k,d)$, the maximum possible number of subsets of $[n]$ of size $d$ that can be shattered by a family of size $k$. Among other results, we determine $f(n,k,d)$ exactly for $d \leq 2$ and show that if $d$ and $n$ grow, with both $d$ and $n-d$ tending to infinity, then, for any $k$ satisfying $2^d \leq k \leq (1+o(1))2^d$, we have $f(n,k,d)=(1+o(1))c\binom{n}{d}$, where $c$, roughly $0.289$, is the probability that a large square matrix over $\mathbb{F}_2$ is invertible. This latter result extends work of Das and M\'esz\'aros. As an application, we improve bounds for the existence of covering arrays for certain alphabet sizes.