AGT: On the intersection density of transitive groups with degree 3p

Talk by Sarobidy Razafimahatratra.

Given a finite transitive group G ≤ Sym(Ω), a subset 𝓕 ⊆ G is intersecting if any two elements of 𝓕 agree on some elements of Ω. The intersection density of G is the rational number ρ(G) given by the maximum ratio |𝓕| / (|G| / |Ω|), where 𝓕 runs through all intersecting sets of G.

Most results on the intersection density focus on particular families of transitive groups. One can look at problems on the intersection density from another perspective. Given an integer n ≥ 3, we would like to determine the possible intersection densities of transitive groups of degree n. This problem turns out to be extremely difficult even in the case where n is a product of two primes.

In 2022, Meagher asked whether ρ(G) ∈ {1, 3/2, 3} for any transitive group G ≤ Sym(Ω) of degree |Ω| = 3p, where p ≥ 5 is an odd prime.

In this talk, I will present some recent progress on this question. I will also talk about more general results in the case where n is a product of two distinct odd primes.