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

TL;DR

This paper investigates the intersection density of symmetric and alternating groups acting on small subsets, proving it equals 1 for subsets of size 3, 4, and 5 using representation theory and ratio bounds.

Contribution

It establishes the exact intersection density for symmetric and alternating groups acting on small subsets, a result not previously known for these actions.

Findings

Intersection density is 1 for Sym(n) acting on 3, 4, 5-subsets.

Uses representation theory and ratio bounds to prove the result.

Provides new insights into the combinatorial structure of these group actions.

Abstract

Given a finite transitive group $G\leq \operatorname{Sym}(\Omega)$, 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 $\operatorname{Sym}(n)$ or $\operatorname{Alt}(n)$ acting on the $k$-subsets of $\{1,2,3\ldots,n\}$, for $k\in \{3,4,5\}$, then $\rho(G)=1$. Our proof relies on the representation theory of the symmetric group and the ratio bound.