On Lagrangians of $3$-uniform hypergraphs

Abstract

Frankl and Füredi conjectured in 1989 that the maximum Lagrangian of all $r$-uniform hypergraphs of fixed size $m$ is realized by the minimum hypergraph $C_{r,m}$ under the colexicographic order. In this paper, we prove a weaker version of the Frankl and Füredi's conjecture at $r=3$: there exists an absolute constant $c>0$ such that for any $3$-uniform hypergraph $H$ with $m$ edges, the Lagrangian of $H$ satisfies $\lambda (H)\le \lambda (C_{3,m+c{m}^{2/9}})$. In particular, this result implies that the Frankl and Füredi's conjecture holds for $r=3$ and $m\in [(\genfrac{}{}{0ex}{}{t-1}{3}),(\genfrac{}{}{0ex}{}{t}{3})-(t-2)-c{t}^{\frac{2}{3}}]$. It improves a recent result of Tyomkyn.

9 page, 1 figure

References in corpus (3)

• Lagrangians of hypergraphs: The Frankl-Füredi conjecture holds almost everywhere
• The maximum $p$-Spectral Radius of Hypergraphs with $m$ Edges
• Symmetric functions and the principal case of the Frankl-Füredi conjecture

Cited by in corpus (2)

• On Hypergraph Lagrangians and Frankl-Füredi's Conjecture
• Lagrangian densities of short 3-uniform linear paths and Turán numbers of their extensions