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 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 Füredi'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=tr−p where 0≤p≤t−r under some conditions. As an implication, we also derive that Frankl and Füredi's conjecture holds for 3-uniform graphs with m=t3−p edges where 0≤p≤4.