Given a finite transitive group ▫$G\leq \sym(\Omega)$▫, a subset ▫$\mathcal{F}$▫ of ▫$G$▫ is \emph{intersecting} if any two elements of ▫$\mathcal{F}$▫ agree on some elements 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 ▫$\sym(n)$▫ or ▫$\alt(n)$▫ 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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