Stability theorems for forbidden configurations

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) $\mathrm{forb}(m,F)=Θ(m^2)$ and the size of $A\in \mathrm{Avoid}(m,F)$ deviates from $\mathrm{forb}(m,F)$ by a linear amount, or (iii) $\mathrm{forb}(m,F)=Θ(m)$ and the size of $A$ is smaller by a constant, then the structure of $A$ is same as the structure of a matrix in $\mathrm{ext}(m,F)$.