Dense 3-uniform hypergraphs containing a large clique

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 Turan 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 Turan 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.