**
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