Lagrangian densities of hypergraph cycles

Abstract: The Lagrangian density of an $r$-uniform hypergraph $F$ is $r!$ multiplying the supremum of the Lagrangians of all $F$-free $r$-uniform hypergraphs. For an $r$-graph $H$ with $t$ vertices, it is clear that $\pi_{\lambda}(H)\ge r!\lambda{(K_{t-1}^r)}$. We say that an $r$-unform hypergraph $H$ with $t$ vertices is perfect if $\pi_{\lambda}(H)= r!\lambda{(K_{t-1}^r)}$. A theorem of Motzkin-Straus implies that all $2$-uniform graphs are perfect. It is interesting to explore what kind of hypergraphs are perfect. A hypergraph is linear if any 2 edges have at most 1 vertex in common. We propose the following conjecture: (1) For $r\ge 3$, there exists $n$ such that a linear $r$-unofrm hypergraph with at least $n$ vertices is perfect. (2) For $r\ge 3$, there exists $n$ such that if $G, H$ are perfect $r$-uniform hypergraphs with at least $n$ vertices, then $G\bigsqcup H$ is perfect. Regarding this conjecture, we obtain a partial result: Let $S_{2,t}={123,124,125,126,...,12(t+2)}$. (An earlier result of Sidorenko states that $S_{2,t}$ is perfect \cite{Sidorenko-89}.) Let $H$ be a perfect $3$-graph with $s$ vertices. Then $F=S_{2,t}\bigsqcup H$ is perfect if $s\geq 3$ and $t\geq 3$.