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{\bf Bal\'azs Patk\'os}
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{\bf How Different Can Two Intersecting Families Be?}
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To measure the difference between two intersecting families ${\cal
F},{\cal G} \subseteq 2^{[n]}$ we introduce the quantity $D({\cal F}
,{\cal G})=|\{ (F,G):F \in {\cal F}, G \in {\cal G}, F\cap G
=\emptyset \}|$. We prove that if ${\cal F}$ is $k$-uniform and ${\cal
G}$ is $l$-uniform, then for large enough $n$ and for any $i \neq j$
${\cal F}_i=\{F \subseteq [n]: i \in F, |F|=k \}$ and ${\cal F}_j=\{F
\subseteq [n]: j \in F, |F|=l\}$ form an optimal pair of families
(that is $D({\cal F},{\cal G}) \leq D({\cal F}_i,{\cal F}_j)$ for all
uniform and intersecting ${\cal F}$ and ${\cal G}$), while in the
non-uniform case any pair of the form ${\cal F}_i=\{F \subseteq [n]: i
\in F \}$ and ${\cal F}_j=\{F \subseteq [n]: j \in F\}$ is optimal for
any $n$.

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