Better upper bounds on the F\"uredi-Hajnal limits of permutations

TL;DR

This paper improves upper bounds on the F"uredi-Hajnal limits for permutation matrices, providing tighter asymptotic bounds for random and all permutation matrices, and extends results to higher-dimensional cases.

Contribution

It establishes new upper bounds on the F"uredi-Hajnal constants for permutation matrices, including random and worst-case scenarios, and generalizes the problem to higher dimensions.

Findings

For random k-permutation matrices, c_P  2^{O(k^{2/3} ext{log}^{7/3}k / ( ext{log} ext{log}k)^{1/3})}.

For all k-permutation matrices, c_P  2^{(4+o(1))k}.

Upper bounds on c_P in terms of Stanley-Wilf limit s_P: c_P  O(s_P^{2.75} ext{log} s_P).

Abstract

A binary matrix is a matrix with entries from the set $\{0,1\}$. We say that a binary matrix $A$ contains a binary matrix $S$ if $S$ can be obtained from $A$ by removal of some rows, some columns, and changing some $1$-entries to $0$-entries. If $A$ does not contain $S$, we say that $A$ avoids $S$. A $k$-permutation matrix $P$ is a binary $k \times k$ matrix with exactly one $1$-entry in every row and one $1$-entry in every column. The F\"uredi-Hajnal conjecture, proved by Marcus and Tardos, states that for every permutation matrix $P$, there is a constant $c_P$ such that for every $n \in \mathbb{N}$, every $n \times n$ binary matrix $A$ with at least $c_P n$ $1$-entries contains $P$. We show that $c_P \le 2^{O(k^{2/3}\log^{7/3}k / (\log\log k)^{1/3})}$ asymptotically almost surely for a random $k$-permutation matrix $P$. We also show that $c_P \le 2^{(4+o(1))k}$ for every $k$-permutation matrix $P$, improving the constant in the exponent of a recent upper bound on $c_P$ by Fox. Moreover, we improve the upper bound on $c_P$ in terms of the Stanley-Wilf limit $s_P$ to $c_P \le O\big(s_P^{2.75} \log s_P\big)$. We also consider a higher-dimensional generalization of the Stanley-Wilf conjecture about the number of $d$-dimensional $n$-permutation matrices avoiding a fixed $d$-dimensional $k$-permutation matrix, and prove almost matching upper and lower bounds of the form $(2^k)^{O(n)} \cdot (n!)^{d-1-1/(d-1)}$ and $n^{-O(k)} k^{\Omega(n)} \cdot (n!)^{d-1-1/(d-1)}$, respectively.