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arXiv · 2201.09727

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

Angelot Behajaina·Roghayeh Maleki·Andriaherimanana Sarobidy Razafimahatratra
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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.

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Angelot Behajaina, Roghayeh Maleki, Andriaherimanana Sarobidy Razafimahatratra. 2022-01-24. On the intersection density of the symmetric group acting on uniform subsets of small size. https://arxiv.org/abs/2201.09727

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