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

Abstract

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)=limn→∞ex(n,F)(rn)\pi \left( F \right) = {\lim _{n \to \infty }}{{ex\left( {n,F} \right)} \over {\left( {_r^n} \right)}}. The Lagrangian density of F is π (F ) = sup{r!λ(G) : G is F -free}, where λ(G) is the Lagrangian of G. Sidorenko observed that π(F ) ≤ π (F ), and Pikhurko observed that π(F ) = π (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 K53K_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
• 05c65