On the Largest Graph-Lagrangian of 3-Graphs with Fixed Number of Edges.

The Graph-Lagrangian of a hypergraph has been a useful tool in hypergraph extremal problems. In most applications, we need an upper bound for the Graph-Lagrangian of a hypergraph. Frankl and Füredi conjectured that the $${r}$$ -graph with $$m$$ edges formed by taking the first $$\textit{m}$$ sets in the colex ordering of the collection of all subsets of $${\mathbb N}$$ of size $${r}$$ has the largest Graph-Lagrangian of all $$r$$ -graphs with $$m$$ edges. In this paper, we show that the largest Graph-Lagrangian of a class of left-compressed $$3$$ -graphs with $$m$$ edges is at most the Graph-Lagrangian of the $$\mathrm 3 $$ -graph with $$m$$ edges formed by taking the first $$m$$ sets in the colex ordering of the collection of all subsets of $${\mathbb N}$$ of size $${3}$$ . [ABSTRACT FROM AUTHOR]