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 $\text{CF}$ is $r$-wise $t$-intersecting if for any $F_1,\dots ,F_r\in \text{CF}$, we have $\text{abs}{\cap }_{i=1}^r{F}_i\text{gs}t$. An $r$-wise $t$-intersecting family of $r+1$ sets $\{T_1,\dots ,T_{r+1}\}$ is called an $(r+1,t)$-triangle if $\mathrm{\mid }{T}_1\cap \cdots \cap T_{r+1}\mathrm{\mid }\text{ls}t-1$. In this paper, we prove that if $n\text{gs}{n}_0(r,t,k)$, then the $r$-wise $t$-intersecting family $\text{CF}\subseteq \left(\genfrac{}{}{0px}{}{[n]}{k}\right)$ containing the most $(r+1,t)$-triangles is isomorphic to $\text{curlybraces}F\in \left(\genfrac{}{}{0px}{}{[n]}{k}\right):\text{abs}F\cap [r+t]\text{gs}r+t-1$. This can also be regarded as a generalized Turán type result.