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 Fredi conjectured that the r r -graph with m m edges formed by taking the first m m sets in the colex ordering of N(r) N ( r ) has the largest Lagrangian of all r r -graphs with m m edges. Talbot in Talbot (2002) provided some evidences for Frankl and Fredis conjecture. In this paper, we prove that the r r -graph with m m edges formed by taking the first m m sets in the colex ordering of N(r) N ( r ) has the largest Lagrangian of all r r -uniform graphs on t t vertices with m m edges when m = t r ? p where 0?p?t?r 0 ? p ? t ? r under some conditions. As an implication, we also derive that Frankl and Fredis conjecture holds for 3-uniform graphs with m = t 3 ? p edges where 0?p?4 0 ? p ? 4 .
colex ordering lagrangians of r r -graphs r r r r r extremal problems in hypergraphs