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{\bf Dhruv Mubayi and Yi Zhao}
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{\bf On a Two-Sided Tur\'an Problem}
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Given positive integers $n,k,t$, with $2 \le k\le n$, and $t
<2^k$, let $m(n,k,t)$ be the minimum size of a family ${\cal F}$
of nonempty subsets of $[n]$ such that every $k$-set in $[n]$
contains at least $t$ sets from ${\cal F}$, and every $(k-1)$-set in
$[n]$ contains at most $t-1$ sets from ${\cal F}$. Sloan et al.
determined $m(n, 3, 2)$ and F\"uredi et al. studied $m(n, 4, t)$
for $t=2, 3$. We consider $m(n, 3, t)$ and $m(n, 4, t)$ for all
the remaining values of $t$ and obtain their exact values except
for $k=4$ and $t= 6, 7, 11, 12$. For example, we prove that $ m(n,
4, 5) = {n \choose 2}-17$ for $n\ge 160$. The values of $m(n, 4,
t)$ for $t=7,11,12$ are determined in terms of well-known (and
open) Tur\'an problems for graphs and hypergraphs. We also obtain
bounds of $m(n, 4, 6)$ that differ by absolute constants.

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