On Frankl and Füredi’s conjecture for 3-uniform hypergraphs

Frankl and Füredi in [1] 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. Denote this r -graph by C r , m and the Lagrangian of a hypergraph by λ ( G ). In this paper, we first show that if ⩽ m ⩽( [ t; 3 ]) , G is a left-compressed 3-graph with m edges and on vertex set [ t ], the triple with minimum colex ordering in G c is ( t − 2 − i )( t − 2) t , then λ ( G ) ≤ λ ( C 3, m ). As an implication, the conjecture of Frankl and Füredi is true for ( [ t; 3 ]) - 6 ⩽ m ⩽( [ t; 3 ]) .