Small Forbidden Configurations III

Abstract

The present paper continues the work begun by Anstee, Ferguson, Griggs and Sali on small forbidden configurations. We define a matrix to be simple if it is a (0,1)-matrix with no repeated columns. Let $F$ be a $k\times l$ (0,1)-matrix (the forbidden configuration). Assume $A$ is an $m\times n$ simple matrix which has no submatrix which is a row and column permutation of $F$. We define ${\hbox{forb}}(m,F)$ as the largest $n$, which would depend on $m$ and $F$, so that such an $A$ exists. 'Small' refers to the size of $k$ and in this paper $k=2$. For $p\le q$, we set $F_{pq}$ to be the $2\times (p+q)$ matrix with $p$ $\bigl[{1\atop0}\bigr]$'s and $q$ $\bigl[{0\atop1}\bigr]$'s. We give new exact values: ${\hbox{forb}}(m,F_{0,4})=\lfloor {5m\over2}\rfloor +2$, ${\hbox{forb}}(m,F_{1,4})=\lfloor {11m\over4}\rfloor +1$, ${\hbox{forb}}(m,F_{1,5})=\lfloor {15m\over4}\rfloor +1$, ${\hbox{forb}}(m,F_{2,4})=\lfloor {10m\over3}-{4\over3}\rfloor$ and ${\hbox{forb}}(m,F_{2,5})=4m$ (For ${\hbox{forb}}(m,F_{1,4})$, ${\hbox{forb}}(m,F_{1,5})$ we obtain equality only for certain classes modulo 4). In addition we provide a surprising construction which shows ${\hbox{forb}}(m,F_{pq})\ge \bigl({p+q\over2}+O(1)\bigr)m$.