DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

Article in press


Authors:

P. Chen, J. Liang and Y. Peng

Title:

The Lagrangian density of $\{123,234,456\}$ and the Turán number of its extension

Source:

Discussiones Mathematicae Graph Theory

Received: 2017-08-04, Revised: 2019-03-18, Accepted: 2019-03-18, https://doi.org/10.7151/dmgt.2219

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 $\pi(F)=\lim_{n\rightarrow\infty} { ex(n,F) \over {n \choose r } }.$ The Lagrangian density of $F$ is $\pi_{\lambda}(F)=\sup \{r! \lambda(G): G$ is $F\hbox{-free}\},$ where $\lambda(G)$ is the Lagrangian of $G$. Sidorenko observed that $\pi(F)\le \pi_{\lambda}(F)$, and Pikhurko observed that $\pi(F)=\pi_{\lambda}(F)$ 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 hypergraph is the balanced complete $5$-partite $3$-uniform hypergraph on $n$ vertices.

Keywords:

Turán number, hypergraph Lagrangian, Lagrangian density

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