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 Tur\'{a}n type result.