Science China Mathematics

, Volume 61, Issue 3, pp 577–592 | Cite as

Dense 3-uniform hypergraphs containing a large clique

  • Biao Wu
  • Yuejian Peng


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.


dense hypergraphs Lagrangian of hypergraphs Turán density 


05C65 05D05 


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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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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.College of Mathematics and EconometricsHunan UniversityChangshaChina
  2. 2.Institute of MathematicsHunan UniversityChangshaChina

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