Repeated Columns and an Old Chestnut

Abstract

Let $t\ge 1$ be a given integer. Let ${\cal F}$ be a family of subsets of $[m]=\{1,2,\ldots ,m\}$. Assume that for every pair of disjoint sets $S,T\subset [m]$ with $|S|=|T|=k$, there do not exist $2t$ sets in ${\cal F}$ where $t$ subsets of ${\cal F}$ contain $S$ and are disjoint from $T$ and $t$ subsets of ${\cal F}$ contain $T$ and are disjoint from $S$. We show that $|{\cal F}|$ is $O(m^{k})$.

Our main new ingredient is allowing, during the inductive proof, multisets of subsets of $[m]$ where the multiplicity of a given set is bounded by $t-1$. We use a strong stability result of Anstee and Keevash. This is further evidence for a conjecture of Anstee and Sali. These problems can be stated in the language of matrices. Let $t\cdot M$ denote $t$ copies of the matrix $M$ concatenated together. We have established the conjecture for those configurations $t\cdot F$ for any $k\times 2$ (0,1)-matrix $F$.