Department Colloquium

Location: CW 113

Speaker: Sarobidy Razafimahatratra, Carleton University

Title:  Erdős-Ko-Rado type theorems for transitive groups of degree a product of two odd primes

Abstract:

Given a finite transitive group \( G\leq \operatorname{Sym}(\Omega)\), a set \(\mathcal{F}\subset G\) is intersecting if for any \( g,h\in G \), there exists \( \omega\in \Omega\) such that \(\omega^g = \omega^h\). The intersection density  \(\rho(G)\) is the maximum ratio of \(\frac{|\mathcal{F}|}{|G_\omega|}\), where \(\mathcal{F}\) runs through all intersecting sets of \(G\) and \(G_\omega\) is the stabilizer of \(\omega\in \Omega\) 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 Marušič 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 character theory of groups 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 a joint work with Angelot Behajaina and Roghayeh Maleki.