05C50 Online Seminar: Sarobidy Razafimahatratra

Given a finite transitive group G ≤ Sym(Ω), a set F ⊂ G is intersecting if for any g, h ∈ G, there exists ω ∈ Ω such that ω g = ω h . The intersection den- sity ρ(G) is the maximum ratio of |F| |Gω| , where F runs through all intersecting sets of G and Gω is the stabilizer of ω ∈ Ω in G. It was conjectured by Meagher et al. in [“On triangles in derangement graphs”, J. Combin. Theory Ser. A, 180:105390, 2021] that any transitive group of degree a product of two distinct odd primes has intersection density equal to 1. This conjecture was disproved by Maruˇsiˇc et al. in [“On intersection density of transitive groups of degree a product of two odd primes.” Finite Fields Appl., 78, 101975,2022] by constructing a family of imprimitive groups of degree pq, where p > q are odd primes, with intersection density equal to q. Therefore, it is natural to ask whether one can classify all possible intersection densities of transitive groups of degree a product of two distinct odd primes. In this talk, I will present some recent developments on this problem. In particular, I will use linear algebra techniques to deal with most quasiprimitive cases. I will then show that there is a deep connection between the non- quasiprimitive cases and full-weight-free cyclic codes. This is based on joint work with Angelot Behajaina and Roghayeh Maleki.