On the intersection density of the symmetric group acting on uniform subsets of small size

Abstract

Given a finite transitive group $G\le \mathrm{Sym}(\Omega )$, a subset $\mathcal{F}$ of $G$ is \emph{intersecting} if any two elements of $\mathcal{F}$ agree on some element of $\Omega$. The \emph{intersection density} of $G$, denoted by $\rho (G)$, is the maximum of the rational number $|\mathcal{F}|{\left(\frac{|G|}{|\Omega |}\right)}^{-1}$ when $\mathcal{F}$ runs through all intersecting sets in $G$. In this paper, we prove that if $G$ is the group $\mathrm{Sym}(n)$ or $\mathrm{Alt}(n)$ acting on the $k$-subsets of $\{1,2,3\dots ,n\}$, for $k\in \{3,4,5\}$, then $\rho (G)=1$. Our proof relies on the representation theory of the symmetric group and the ratio bound.

31 pages