Lower bound on the size of a quasirandom forcing set of permutations

TL;DR

This paper establishes a fundamental lower bound of four on the size of any forcing set of permutations, advancing understanding of quasirandom permutation structures.

Contribution

It provides the first non-trivial lower bound on the size of forcing sets of permutations, showing they must contain at least four permutations.

Findings

Any forcing set of permutations has at least four elements.

The result applies to permutations of arbitrary orders.

It advances the theoretical understanding of quasirandom permutation sets.

Abstract

A set $S$ of permutations is forcing if for any sequence $\{\Pi_i\}_{i \in \mathbb{N}}$ of permutations where the density $d(\pi,\Pi_i)$ converges to $\frac{1}{|\pi|!}$ for every permutation $\pi \in S$, it holds that $\{\Pi_i\}_{i \in \mathbb{N}}$ is quasirandom. Graham asked whether there exists an integer $k$ such that the set of all permutations of order $k$ is forcing; this has been shown to be true for any $k\ge 4$. In particular, the set of all twenty-four permutations of order $4$ is forcing. We provide the first non-trivial lower bound on the size of a forcing set of permutations: every forcing set of permutations (with arbitrary orders) contains at least four permutations.