Symmetric functions and the principal case of the Frankl-F\"uredi conjecture

TL;DR

This paper proves bounds on a symmetric function related to hypergraph edge weights, extending the Frankl-F"uredi conjecture for certain cases using new bounds on elementary symmetric functions.

Contribution

It establishes new bounds on the maximum of symmetric functions for hypergraphs when certain parameters are met, advancing the understanding of the Frankl-F"uredi conjecture.

Findings

Proves bounds for hypergraphs with r ≤ 5 or large t values.

Characterizes cases of equality when t is an integer.

Introduces new bounds on elementary symmetric functions.

Abstract

Let $r\geq3$ and $G$ be an $r$-uniform hypergraph with vertex set $\left\{ 1,\ldots,n\right\} $ and edge set $E$. Let \[ \mu\left( G\right) :=\max {\textstyle\sum\limits_{\left\{ i_{1},\ldots,i_{r}\right\} \in E}} x_{i_{1}}\cdots x_{i_{r}}, \] where the maximum is taken over all nonnegative $x_{1},\ldots,x_{n}$ with $x_{1}+\cdots+x_{n}=1.$ Let $t\geq r-1$ be the unique real number such that $\left\vert E\right\vert =\binom{t}{r}$. It is shown that if $r\leq5$ or $t\geq4\left( r-1\right) \left( r-2\right) $, then \[ \mu\left( G\right) \leq t^{-r}\binom{t}{r}% \] with equality holding if and only if $t$ is an integer. The proof is based on some new bounds on elementary symmetric functions.