Stability Theorems for Forbidden Configurations

TL;DR

This paper investigates stability phenomena in forbidden configurations within (0,1)-matrices, establishing conditions under which matrices close to extremal size maintain a specific structure.

Contribution

It provides new stability theorems for forbidden configurations, characterizing when matrices near extremal size resemble extremal structures in various growth regimes.

Findings

Matrices with size close to extremal have similar structure.

Characterization of forbidden configurations with quadratic and linear growth.

Structural stability results for different asymptotic regimes.

Abstract

Stability is a well investigated concept in extremal combinatorics. The main idea is that if some object is close in size to an extremal object, then it retains the structure of the extremal construction. In the present paper we study stability in the context of forbidden configurations. $(0,1)$-matrix $F$ is a configuration in a $(0,1)$-matrix $A$ if $F$ is a row and columns permutation of a submatrix of $A$. $\mathrm{Avoid}(m,F)$ denotes the set of $m$-rowed $(0,1)$-matrices with pairwise distinct columns without configuration $F$, $\mathrm{forb}(m,F)$ is the largest number of columns of a matrix in $\mathrm{Avoid}(m,F)$, while $\mathrm{ext}(m,F)$ is the set of matrices in $\mathrm{Avoid}(m,F)$ of size $\mathrm{forb}(m,F)$. We show cases (i) when each element of $\mathrm{Avoid}(m,F)$ have the structure of element(s) in $\mathrm{ext}(m,F)$, (ii) $\mathrm{forb}(m,F)=\Theta(m^2)$ and the size of $A\in \mathrm{Avoid}(m,F)$ deviates from $\mathrm{forb}(m,F)$ by a linear amount, or (iii) $\mathrm{forb}(m,F)=\Theta(m)$ and the size of $A$ is smaller by a constant, then the structure of $A$ is same as the structure of a matrix in $\mathrm{ext}(m,F)$.