Tur\'an number of special four cycles in triple systems

A {\em special four-cycle } $F$ in a triple system consists of four triples {\em inducing } a $C_4$. This means that $F$ has four special vertices $v_1,v_2,v_3,v_4$ and four triples in the form $w_iv_iv_{i+1}$ (indices are understood $\pmod 4$) where the $w_j$s are not necessarily distinct but disjoint from $\{v_1,v_2,v_3,v_4\}$. There are seven non-isomorphic special four-cycles, their family is denoted by $\cal{F}$. Our main result implies that the Tur\'an number $\text{ex}(n,{\cal{F}})=\Theta(n^{3/2})$. In fact, we prove more, $\text{ex}(n,\{F_1,F_2,F_3\})=\Theta(n^{3/2})$, where the $F_i$-s are specific members of $\cal{F}$. This extends previous bounds for the Tur\'an number of triple systems containing no Berge four cycles. We also study $\text{ex}(n,{\cal{A}})$ for all ${\cal{A}}\subseteq {\cal{F}}$. For 16 choices of $\cal{A}$ we show that $\text{ex}(n,{\cal{A}})=\Theta(n^{3/2})$, for 92 choices of $\cal{A}$ we find that $\text{ex}(n,{\cal{A}})=\Theta(n^2)$ and the other 18 cases remain unsolved.