Connection between the clique number and the Lagrangian of $3$-uniform hypergraphs

TL;DR

This paper investigates the relationship between clique number and Lagrangian in 3-uniform hypergraphs, confirming a conjecture for larger ranges of edges and deriving Turán-type results for specific hypergraph classes.

Contribution

It extends the connection between clique number and Lagrangian to 3-uniform hypergraphs, confirming Furedi and Frankl's conjecture for broader cases and providing Turán-type bounds.

Findings

Confirmed Furedi and Frankl's conjecture for larger m in 3-uniform hypergraphs.

Established weaker Turán-type theorems for left-compressed 3-uniform hypergraphs.

Demonstrated the connection between clique number and Lagrangian in the hypergraph setting.

Abstract

There is a remarkable connection between the clique number and the Lagrangian of a 2-graph proved by Motzkin and Straus in 1965. It is useful in practice if similar results hold for hypergraphs. However the obvious generalization of Motzkin and Straus' result to hypergraphs is false. Frankl and F\"{u}redi conjectured that the $r$-uniform hypergraph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-uniform hypergraphs with $m$ edges. For $r=2$, Motzkin and Straus' theorem confirms this conjecture. For $r=3$, it is shown by Talbot that this conjecture is true when $m$ is in certain ranges. In this paper, we explore the connection between the clique number and Lagrangians for $3$-uniform hypergraphs. As an application of this connection, we confirm that Frankl and F\"{u}redi's conjecture holds for bigger ranges of $m$ when $r$=3. We also obtain two weaker versions of Tur\'{a}n type theorem for left-compressed $3$-uniform hypergraphs.