Some results on Lagrangians of hypergraphs.

The Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. In most applications, we need an upper bound for the Lagrangian of a hypergraph. Frankl and Füredi conjectured that the -graph with edges formed by taking the first sets in the colex ordering of has the largest Lagrangian of all -graphs with edges. Talbot in Talbot (2002) provided some evidences for Frankl and Füredi’s conjecture. In this paper, we prove that the -graph with edges formed by taking the first sets in the colex ordering of has the largest Lagrangian of all -uniform graphs on vertices with edges when where under some conditions. As an implication, we also derive that Frankl and Füredi’s conjecture holds for 3-uniform graphs with edges where . [ABSTRACT FROM AUTHOR]