The Tur\'an number of Berge-K_4 in triple systems

A Berge-$K_4$ in a triple system is a configuration with four vertices $v_1,v_2,v_3,v_4$ and six distinct triples $\{e_{ij}:1\le i<j\le 4\}$ such that $\{v_i,v_j\}\subset e_{ij}$ for every $1\le i<j\le 4$. We denote by $\mathcal{B}$ the set of Berge-$K_4$ configurations. A triple system is $\mathcal{B}$-free if it does not contain any member of $\mathcal{B}$. We prove that the maximum number of triples in a $\mathcal{B}$-free triple system on $n\ge 6$ points is obtained by the balanced complete $3$-partite triple system: all triples $\{abc:a\in A,b\in B,c\in C\}$ where $A,B,C$ is a partition of $n$ points with $$\lfloor \frac{n}{3}\rfloor =\mathrm{\mid }A\mathrm{\mid }\le \mathrm{\mid }B\mathrm{\mid }\le \mathrm{\mid }C\mathrm{\mid }=\lceil \frac{n}{3}\rceil \mathrm{.}$$