# Dense 3-uniform hypergraphs containing a large clique

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## Abstract

An *r*-uniform graph *G* is dense if and only if every proper subgraph *G*′ of G satisfies *λ*(*G*′) < *λ*(*G*), where *λ*(*G*) is the Lagrangian of a hypergraph *G*. In 1980’s, Sidorenko showed that *π*(*F*), the Turán density of an *r*-uniform hypergraph *F* is *r*! multiplying the supremum of the Lagrangians of all dense *F*-hom-free *r*-uniform hypergraphs. This connection has been applied in the estimating Turán density of hypergraphs. When *r* = 2, the result of Motzkin and Straus shows that a graph is dense if and only if it is a complete graph. However, when *r* ≥ 3, it becomes much harder to estimate the Lagrangians of *r*-uniform hypergraphs and to characterize the structure of all dense *r*-uniform graphs. The main goal of this note is to give some sufficient conditions for 3-uniform graphs with given substructures to be dense. For example, if G is a 3-graph with vertex set [*t*]and *m* edges containing [*t* − 1]^{(3)}, then *G* is dense if and only if \(m \geqslant \left( {\frac{{t - 1}}{3}} \right) + \left( {\frac{{t - 2}}{2}} \right) + 1\)
. We also give a sufficient condition on the number of edges for a 3-uniform hypergraph containing a large clique minus 1 or 2 edges to be dense.

### Keywords

dense hypergraphs Lagrangian of hypergraphs Turán density### MSC(2010)

05C65 05D05## Preview

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## Notes

### Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11271116). The authors thank both reviewers for reading the manuscript carefully, checking all the details and giving insightful comments to help improve the manuscript.

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