Bounding extremal functions of forbidden $0-1$ matrices using $(r,s)$-formations

TL;DR

This paper establishes tight bounds on the extremal functions of certain forbidden ordered sequences and 0-1 matrices using the concept of $(r,s)$-formations, advancing understanding in combinatorial matrix theory.

Contribution

It introduces a method to derive tight bounds on extremal functions for forbidden sequences and matrices using $(r,s)$-formations, connecting sequence and matrix extremal problems.

Findings

Derived tight bounds for extremal functions of specific ordered sequences.

Extended the $(r,s)$-formation technique to 0-1 matrix extremal problems.

Provided a unified approach for sequence and matrix extremal bounds.

Abstract

First, we prove tight bounds of $n 2^{\frac{1}{(t-2)!}\alpha(n)^{t-2} \pm O(\alpha(n)^{t-3})}$ on the extremal function of the forbidden pair of ordered sequences $(1 2 3 \ldots k)^t$ and $(k \ldots 3 2 1)^t$ using bounds on a class of sequences called $(r,s)$-formations. Then, we show how an analogous method can be used to derive similar bounds on the extremal functions of forbidden pairs of $0-1$ matrices consisting of horizontal concatenations of identical identity matrices and their horizontal reflections.