Symmetric functions and the principal case of the Frankl-Füredi conjecture

Abstract

Let $r\ge 3$ and $G$ be an $r$-uniform hypergraph with vertex set $\left\{1,\dots ,n\right\}$ and edge set $E$. Let \[ μ\left( G\right) :=\max {\textstyle\sum\limits_{\left\{ i_{1},\ldots,i_{r}\right\} \in E}} x_{i_{1}}\cdots x_{i_{r}}, \] where the maximum is taken over all nonnegative $x_1,\dots ,x_n$ with $x_1+\cdots +x_n=1.$ Let $t\ge r-1$ be the unique real number such that $\left|E\right|=\left(\genfrac{}{}{0ex}{}{t}{r}\right)$. It is shown that if $r\le 5$ or $t\ge 4\left(r-1\right)\left(r-2\right)$, then \[ μ\left( G\right) \leq t^{-r}\binom{t}{r}% \] with equality holding if and only if $t$ is an integer. The proof is based on some new bounds on elementary symmetric functions.

In v2 and v3, the main result is extended; 15 pages

References in corpus (1)

• Lagrangians of hypergraphs: The Frankl-Füredi conjecture holds almost everywhere

Cited by in corpus (3)

• On Lagrangians of $3$-uniform hypergraphs
• The maximum $p$-Spectral Radius of Hypergraphs with $m$ Edges
• On Hypergraph Lagrangians and Frankl-Füredi's Conjecture