Maximum shattering

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 Mészáros. As an application, we improve bounds for the existence of covering arrays for certain alphabet sizes.