Triangles in r-wise t-intersecting families

Let $t$, $r$, $k$ and $n$ be positive integers and $\mathcal{F}$ a family of $k$-subsets of an $n$-set $V$. The family $ \CF $ is $r$-wise $t$-intersecting if for any $ F_1, \ldots, F_r \in \CF $, we have $ \abs{\cap_{i = 1}^{r}F_i}\gs t $. An $r$-wise $t$-intersecting family of $r + 1$ sets $\{T_1, \ldots, T_{r + 1}\}$ is called an $(r + 1,t)$-triangle if $ |T_1 \cap \cdots \cap T_{r + 1}| \ls t - 1 $. In this paper, we prove that if $ n \gs n_0(r, t, k) $, then the $r$-wise $t$-intersecting family $ \CF \subseteq \binom{[n]}{k} $ containing the most $(r + 1,t)$-triangles is isomorphic to $ \curlybraces{F \in \binom{[n]}{k}: \abs{F \cap [r + t]} \gs r + t - 1} $. This can also be regarded as a generalized Turán type result.

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