Maximum Shattering

Abstract

A family $ℱ$ of subsets of $\left[n\right]=\left\{1,2,\dots ,n\right\}$ shatters a set $A\subseteq \left[n\right]$ if for every $A^{\prime }\subseteq A$, there is an $F\in ℱ$ such that $F\cap A=A^{\text{'}}$. We develop a framework to analyze $f\left(n,k,d\right)$, the maximum possible number of subsets of $\left[n\right]$ of size $d$ that can be shattered by a family of size $k$. Among other results, we determine $f\left(n,k,d\right)$ exactly for $d\le 2$ and show that if $d$ and $n$ grow, with both $d$ and $n-d$ tending to infinity, then for any $k$ satisfying $2^d\le k\le \left(1+o\left(1\right)\right)2^d$, we have $f\left(n,k,d\right)=\left(1+o\left(1\right)\right)c\left(\frac{n}{d}\right)$, where $c$, roughly 0.289, is the probability that a large square matrix over $F_2$ 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.