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

Given a finite transitive group G <= Sym(12), a subset F of G is intersecting if any two elements of F agree on some elements of 12. The intersection density of G, denoted by rho(G), is the )-1 maximum of the rational number |F| (|G|when F runs |omega| 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 ... , n}, for k is an element of {3, 4, 5}, then rho(G) = 1. Our proof relies on the representation theory of the symmetric group and the ratio bound. (c) 2023 Elsevier Inc. All rights reserved.