Forbidden formations in 0-1 matrices

TL;DR

This paper extends bounds on the extremal function of forbidden 0-1 matrices from two dimensions to multiple dimensions, introducing new pattern families and proving tight bounds for many cases.

Contribution

It generalizes extremal function bounds to multidimensional matrices and introduces the $(P, s)$-formation pattern family with tight bounds in key cases.

Findings

Extremal function for multidimensional matrices is $O(n^{d-1}2^{eta(n)^{t}})$.

$(P, s)$-formations generalize previous pattern families.

Tight bounds are established for permutation matrices with at least two ones.

Abstract

Keszegh (2009) proved that the extremal function $ex(n, P)$ of any forbidden light $2$-dimensional 0-1 matrix $P$ is at most quasilinear in $n$, using a reduction to generalized Davenport-Schinzel sequences. We extend this result to multidimensional matrices by proving that any light $d$-dimensional 0-1 matrix $P$ has extremal function $ex(n, P,d) = O(n^{d-1}2^{\alpha(n)^{t}})$ for some constant $t$ that depends on $P$. To prove this result, we introduce a new family of patterns called $(P, s)$-formations, which are a generalization of $(r, s)$-formations, and we prove upper bounds on their extremal functions. In many cases, including permutation matrices $P$ with at least two ones, we are able to show that our $(P, s)$-formation upper bounds are tight.