Small families of partially shattering permutations

TL;DR

This paper investigates the minimum size of permutation families that partially shatter all subsets of a certain size, revealing new regimes of behavior and disproving a long-standing conjecture for larger subset sizes.

Contribution

It introduces a new regime for the function $f_k(n,t)$ when $k \,\geq\, 4$, showing that partial shattering can behave differently than previously thought.

Findings

Established that $f_k(n,t) = \Theta(\sqrt{\log n})$ for certain $t$ when $k \ge 4$

Proved that $f_k(n,t) = \Theta(\log n)$ for $t > 2^{k-1}$

Narrowed the unknown range of $t$ for the asymptotic behavior of $f_k(n,t)$

Abstract

We say that a family of permutations $t$-shatters a set if it induces at least $t$ distinct permutations on that set. What is the minimum number $f_k(n,t)$ of permutations of $\{1, \dots, n\}$ that $t$-shatter all subsets of size $k$? For $t \le 2$, $f_k(n,t) = \Theta(1)$. Spencer showed that $f_k(n,t) = \Theta(\log \log n)$ for $3 \le t \le k$ and $f_k(n,k!) = \Theta(\log n)$. In 1996, F\"uredi asked whether partial shattering with permutations must always fall into one of these three regimes. Johnson and Wickes recently settled the case $k = 3$ affirmatively and proved that $f_k(n,t) = \Theta(\log n)$ for $t > 2 (k-1)!$. We give a surprising negative answer to the question of F\"uredi by showing that a fourth regime exists for $k \ge 4$. We establish that $f_k(n,t) = \Theta(\sqrt{\log n})$ for certain values of $t$ and prove that this is the only other regime when $k = 4$. We also show that $f_k(n,t) = \Theta(\log n)$ for $t > 2^{k-1}$. This greatly narrows the range of $t$ for which the asymptotic behaviour of $f_k(n,t)$ is unknown.