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We investigate extremal problems for hypergraphs satisfying the followingdensity condition. A $3$-uniform hypergraph $H=(V, E)$ is $(d, \eta,P_2)$-denseif 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 $\mathcal{K}_{P_2}(P,Q)$ denotesthe 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 thatfor sufficiently small $\eta$ every sufficiently large $(d, \eta,P_2)$-densehypergraph $H$ 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 acomplete 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].

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