对称函数与Frankl-Füredi猜想的主要情形
Symmetric functions and the principal case of the Frankl-Füredi conjecture
摘要：
设$r\geq3$和$G$是具有点集$\left\{ 1,\ldots,n\right\} $和边集$E$的$r$ -一致超图.设\[ μ\left( G\right) :=\max {\textstyle\sum\limits_{\left\{ i_{1},\ldots,i_{r}\right\} \in E}} x_{i_{1}}\cdots x_{i_{r}}, \]，其中最大值取所有具有$x_{1}+\cdots+x_{n}=1.$的非负$x_{1},\ldots,x_{n}$。设$t\geq r-1$为唯一实数，使得$\left\vert E\right\vert =\binom{t}{r}$。证明了若$r\leq5$或$t\geq4\left( r-1\right) \left( r-2\right) $，则等式成立的\[ μ\left( G\right) \leq t^{-r}\binom{t}{r}% \]当且仅当$t$是整数.证明是基于一些新的界限上的初等对称函数。
英文摘要：
Let $r\geq3$ and $G$ be an $r$-uniform hypergraph with vertex set $\left\{ 1,\ldots,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 $x_{1},\ldots,x_{n}$ 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 \[ μ\left( G\right) \leq t^{-r}\binom{t}{r}% \] with equality holding if and only if $t$ is an integer. The proof is based on some new bounds on elementary symmetric functions.
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