On Frankl and Furedi's conjecture for 3-uniform hypergraphs

TL;DR

This paper investigates Frankl and Furedi's conjecture on the maximum Lagrangian of 3-uniform hypergraphs with a given number of edges, providing partial results towards its proof.

Contribution

It offers new partial results supporting Frankl and Furedi's conjecture for 3-uniform hypergraphs, advancing understanding in hypergraph extremal problems.

Findings

Partial validation of the conjecture for specific cases

Identification of conditions where the conjecture holds

Progress towards a general proof

Abstract

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 Furedi in \cite{FF} conjectured that the $r$-graph with $m$ edges formed by taking the first $m$ sets in the colex ordering of ${\mathbb N}^{(r)}$ has the largest Lagrangian of all $r$-graphs with $m$ edges. In this paper, we give some partial results for this conjecture.