Description
Abstract:
We prove the following generalized Tur\'an type result. A
collection $\mathcal{T}$ of $r$ sets is an $r$-triangle if for every
$T_1,T_2,\dots,T_{r-1}\in \mathcal{T}$ we have
$\cap_{i=1}^{r-1}T_i\neq\emptyset$, but $\cap_{T\in \mathcal{T}}T$ is
empty. A family $\mathcal{F}$ of sets is $r$-wise intersecting if for
any $F_1,F_2,\dots,F_r\in \mathcal{F}$ we have $\cap_{i=1}^rF_i\neq
\emptyset$ or equivalently if $\mathcal{F}$ does not contain any
$m$-triangle for $m=2,3,\dots,r$. We prove that if $n\ge n_0(r,k)$, then
the $r$-wise intersecting family $\mathcal{F}\subseteq {[n]\choose k}$
containing the most number of $(r+1)$-triangles is isomorphic to $\{F\in
{[n]\choose k}:|F\cap [r+1]|\ge r\}$.
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