Some remarks on the extremal function for uniformly two-path dense hypergraphs

We investigate extremal problems for hypergraphs satisfying the following density condition. A $3$-uniform hypergraph $H=(V, E)$ is $(d, \eta,P_2)$-dense if for any two subsets of pairs $P$, $Q\subseteq V\times V$ the number of pairs $((x,y),(x,z))\in P\times Q$ with $\{x,y,z\}\in E$ is at least $d|\mathcal{K}_{P_2}(P,Q)|-\eta|V|^3,$ where $\mathcal{K}_{P_2}(P,Q)$ denotes the set of pairs in $P\times Q$ of the form $((x,y),(x,z))$. For a given $3$-uniform hypergraph $F$ we are interested in the infimum $d\geq 0$ such that for sufficiently small $\eta$ every sufficiently large $(d, \eta,P_2)$-dense hypergraph $H$ contains a copy of $F$ and this infimum will be denoted by $\pi_{P_2}(F)$. We present a few results for the case when $F=K_k^{(3)}$ is a complete three uniform hypergraph on $k$ vertices. It will be shown that $\pi_{P_2}(K_{2^r}^{(3)})\leq \frac{r-2}{r-1}$, which is sharp for $r=2,3,4$, where the lower bound for $r=4$ is based on a result of Chung and Graham [Edge-colored complete graphs with precisely colored subgraphs, Combinatorica 3 (3-4), 315-324].