Repeated columns and an old chestnut

Let $t\ge 1$ be a given integer. Let $\mathcal{F}$ be a family of subsets of $[m]=\{1,2,\dots ,m\}$. Assume that for every pair of disjoint sets $S,T\subset [m]$with$|S|=|T|=k$,theredonotexist$2t$setsin${\cal F}$where$t$subsetsof${\cal F}$contain$S$andaredisjointfrom$T$and$t$subsetsof${\cal F}$contain$T$andaredisjointfrom$S$\mathrm{.}Weshowthat$|{\cal F}|$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 $F$.