论文标题
在作用于小尺寸的均匀子集的对称组的交点密度上
On the intersection density of the symmetric group acting on uniform subsets of small size
论文作者
论文摘要
给定一个有限的透明组$ g \ leq \ operatorAtorName {sym}（ω）$，如果$ \ nathcal {f} $的任何两个要素同意$ω$ $ω$的任何两个元素，则$ g $的子集$ \ Mathcal {f} $ is \ emph {Interpecting}。由$ρ（g）$表示的\ emph {交点密度}是有理数$ | \ m rathcal {f} | \ left的最大值（\ frac {| g |} {| g | em} {|ω| | emy} {|ω|} \ right）^{ - right） In this paper, we prove that if $G$ is the group $\operatorname{Sym}(n)$ or $\operatorname{Alt}(n)$ acting on the $k$-subsets of $\{1,2,3\ldots,n\}$, for $k\in \{3,4,5\}$, then $ρ(G)=1$.我们的证明依赖于对称群体的表示理论和比率结合。
Given a finite transitive group $G\leq \operatorname{Sym}(Ω)$, a subset $\mathcal{F}$ of $G$ is \emph{intersecting} if any two elements of $\mathcal{F}$ agree on some element of $Ω$. The \emph{intersection density} of $G$, denoted by $ρ(G)$, is the maximum of the rational number $|\mathcal{F}|\left(\frac{|G|}{|Ω|}\right)^{-1}$ when $\mathcal{F}$ runs through all intersecting sets in $G$. In this paper, we prove that if $G$ is the group $\operatorname{Sym}(n)$ or $\operatorname{Alt}(n)$ acting on the $k$-subsets of $\{1,2,3\ldots,n\}$, for $k\in \{3,4,5\}$, then $ρ(G)=1$. Our proof relies on the representation theory of the symmetric group and the ratio bound.
