# Combinatorics Seminar 2018

Date: Thu, Jul 12, 2018

Time: 09:00 - 10:00

Venue: Middle Meeting Room

Title: Characterization of intersecting families of maximum size in $PSL(2,q)$

Abstract:

The Erdős-Ko-Rado (EKR) theorem is a classical result in extremal set theory. It states that when $k<n/2$, any family of $k$-subsets of an $n$-set $X$, with the property that any two subsets in the family have nonempty intersection, has size at most ${n-1\choose k-1}$; equality holds if and only if the family consists of all $k$-subsets of $X$ containing a fixed point.

Here we consider EKR type problems for permutation groups. In particular, we focus on the action of the $2$-dimensional projective special linear group $PSL(2,q)$ on the projective line $PG(1,q)$ over the finite field ${\mathbb F}_q$, where $q$ is an odd prime power. A subset $S$ of $PSL(2,q)$ is said to be an intersecting family if for any $g_1,g_2 \in S$, there exists an element $x\in PG(1,q)$ such that $x^{g_1}= x^{g_2}$. It is known that the maximum size of an intersecting family in $PSL(2,q)$ is $q(q-1)/2$. We prove that all intersecting families of maximum size are cosets of point stabilizers for all odd prime powers $q>3$.

Joint work with Ling Long, Rafael Plaza, and Peter Sin.

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