Connection between the clique number and the Lagrangian of 3-uniform hypergraphs

There is a remarkable connection between the clique number and the Lagrangian of a 2-graph proved by Motzkin and Straus (J Math 17:533–540, 1965). It would be useful in practice if similar results hold for hypergraphs. However, the obvious generalization of Motzkin and Straus’ result to hypergraphs is false. Frankl and Füredi conjectured that the r-uniform hypergraph with m edges formed by taking the first m sets in the colex ordering of $${\mathbb N}^{(r)}$$N(r) has the largest Lagrangian of all r-uniform hypergraphs with m edges. For $$r=2$$r=2, Motzkin and Straus’ theorem confirms this conjecture. For $$r=3$$r=3, it is shown by Talbot that this conjecture is true when m is in certain ranges. In this paper, we explore the connection between the clique number and Lagrangians for 3-uniform hypergraphs. As an application of this connection, we confirm that Frankl and Füredi’s conjecture holds for bigger ranges of m when $$r=3$$r=3. We also obtain two weaker versions of Turán type theorem for left-compressed 3-uniform hypergraphs.