The Lagrangian Density of {123, 234, 456} and the Turán Number of its Extension

Given a positive integer $n$ and an $r$-uniform hypergraph $F$, the Turán number $ex(n, F)$ is the maximum number of edges in an $F$-free $r$-uniform hypergraph on $n$ vertices. The Turán density of $F$ is defined as \(π(F)=lim_{n→∞}\frac{ex(n,F)}{\binom{n}{r}}\). The Lagrangian density of $F$ is \(π_\lambda(F) = sup\{r!\lambda(G):G\) is \(F-free\}\), where $\lambda(G)$ is the Lagrangian of $G$. Sidorenko observed that \(π(F) ≤ π_\lambda(F)\), and Pikhurko observed that \(π(F) = π_\lambda(F)\) if every pair of vertices in $F$ is contained in an edge of $F$. Recently, Lagrangian densities of hypergraphs and Turán numbers of their extensions have been studied actively. For example, in the paper [A hypergraph Turán theorem via Lagrangians of intersecting families, J. Combin. Theory Ser. A 120 (2013) 2020–2038], Hefetz and Keevash studied the Lagrangian densitiy of the 3-uniform graph spanned by {123, 456} and the Turán number of its extension. In this paper, we show that the Lagrangian density of the 3-uniform graph spanned by {123, 234, 456} achieves only on $K_5^3$. Applying it, we get the Turán number of its extension, and show that the unique extremal hyper-graph is the balanced complete 5-partite 3-uniform hypergraph on $n$ vertices.