Shattering $k$-sets with Permutations

TL;DR

This paper explores the concept of shattering in permutation families, introducing partial shattering variants, analyzing different regimes for small and large k, and establishing bounds on the number of shattered sets by small permutation families.

Contribution

It introduces and studies two natural partial shattering problems for permutations, providing regime classifications and bounds for the size of families needed to shatter sets.

Findings

For k=3, three regimes depending on t: constant, log log n, log n.

Existence of similar regimes for larger k, though not covering all t.

A family of 6 permutations can shatter between 17/42 and 11/14 of all triples.

Abstract

Many concepts from extremal set theory have analogues for families of permutations. This paper is concerned with the notion of shattering for permutations. A family $\mathcal{P}$ of permutations of an $n$-element set $X$ shatters a $k$-set from $X$ if it appears in each of the $k!$ possible orders in some permutation in $\mathcal{P}$. The smallest family $\mathcal{P}$ which shatters every $k$-subset of $X$ is known to have size $\Theta(\log n)$. Our aim is to introduce and study two natural partial versions of this shattering problem. Our first main result concerns the case where our family must contain only $t$ out of $k!$ of the possible orders. When $k=3$ we show that there are three distinct regimes depending on $t$: constant, $\Theta(\log\log n)$, $\Theta(\log n)$. We also show that for larger $k$ these same regimes exist although they may not cover all values of $t$. Our second direction concerns the problem of determining the largest number of $k$-sets that can be totally shattered by a family with given size. We show that for any $n$, a family of $6$ permutations is enough to shatter a proportion between $\frac{17}{42}$ and $\frac{11}{14}$ of all triples.