Title: On the intersection density of transitive groups with degree 3p
| Speaker: | Sarobidy Razafimahatratra |
| Affiliation: | University of Primorska |
| Location: | Please contact Sabrina Lato for Zoom link. |
Abstract: Given a finite transitive group $G\leq \operatorname{Sym}(\Omega)$, a subset $\mathcal{F}\subset G$ is intersecting if any two elements of $\mathcal{F}$ agree on some elements of $\Omega$. The \emph{intersection density} of $G$ is the rational number $\rho(G)$ given by the maximum ratio $\frac{|\mathcal{F}|}{|G|/|\Omega|}$, where $\mathcal{F}$ 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 $n\geq 3$, 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 $\rho(G) \in \left\{1,\frac{3}{2},3\right\}$ for any transitive group $G\leq \operatorname{Sym}(\Omega)$ of degree $|\Omega| = 3p$, where $p\geq 5$ 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