Set Systems and Families of Permutations with Small Traces

TL;DR

This paper investigates the maximum size of set systems and permutation families with small traces, extending Sauer's Lemma and exploring connections to VC-dimension and permutation pattern avoidance.

Contribution

It generalizes Sauer's Lemma to set systems with bounded traces and applies these bounds to families of permutations, linking to VC-dimension and permutation pattern problems.

Findings

Bound on set system size based on trace function

Extension of Sauer's Lemma to permutation families

Connection to VC-dimension and permutation pattern avoidance

Abstract

We study the maximum size of a set system on $n$ elements whose trace on any $b$ elements has size at most $k$. We show that if for some $b \ge i \ge 0$ the shatter function $f_R$ of a set system $([n],R)$ satisfies $f_R(b) < 2^i(b-i+1)$ then $|R| = O(n^i)$; this generalizes Sauer's Lemma on the size of set systems with bounded VC-dimension. We use this bound to delineate the main growth rates for the same problem on families of permutations, where the trace corresponds to the inclusion for permutations. This is related to a question of Raz on families of permutations with bounded VC-dimension that generalizes the Stanley-Wilf conjecture on permutations with excluded patterns.