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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 $O(m^{k})$.\n  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 $t-1$. 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