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 $${r}$$ r -graph with $$m$$ m edges formed by taking the first $$\textit{m}$$ m sets in the colex ordering of the collection of all subsets of $${\mathbb N}$$ N of size $${r}$$ r has the largest Graph-Lagrangian of all $$r$$ r -graphs with $$m$$ m edges. In this paper, we show that the largest Graph-Lagrangian of a class of left-compressed $$3$$ 3 -graphs with $$m$$ m edges is at most the Graph-Lagrangian of the $$\mathrm 3 $$ 3 -graph with $$m$$ m edges formed by taking the first $$m$$ m sets in the colex ordering of the collection of all subsets of $${\mathbb N}$$ N of size $${3}$$ 3 .