Geodesic
The Lagrangian Density of {123, 234, 456} and the Turán Number of its Extension
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 4, pp. 905-921.

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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→∞ex(n,F)/nr. The Lagrangian density of F is π_λ(F) = sup{r!λ(G):G is F-free}, where λ(G) is the Lagrangian of G. Sidorenko observed that π(F) ≤ π_λ(F), and Pikhurko observed that π(F) = π_λ(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.
Keywords: Turán number, hypergraph Lagrangian, Lagrangian density 
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Chen, Pingge; Liang, Jinhua; Peng, Yuejian. The Lagrangian Density of {123, 234, 456} and the Turán Number of its Extension. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 4, pp. 905-921. http://geodesic.mathdoc.fr/item/DMGT_2021_41_4_a3/

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