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 Furedi conjectured that the r -graph with m edges formed by taking the first m sets in the colex ordering of N ( r ) has the largest Lagrangian of all r -graphs with m edges. Talbot in Talbot (2002) provided some evidences for Frankl and Furedi's conjecture. In this paper, we prove that the r -graph with m edges formed by taking the first m sets in the colex ordering of N ( r ) has the largest Lagrangian of all r -uniform graphs on t vertices with m edges when m = t r - p where 0 ? p ? t - r under some conditions. As an implication, we also derive that Frankl and Furedi's conjecture holds for 3-uniform graphs with m = t 3 - p edges where 0 ? p ? 4 .