Erdős-Ko-Rado theorems for ovoidal circle geometries and polynomials over finite fields

Adriaensen, S. (Speaker)

Activity: Talk or presentationTalk or presentation at a workshop/seminar

Description

[Online talk] Given an incidence structure (P, B), we say that a family F contained in B is intersecting if any two elements of F share at least one point. There have been ample investigations into the size and structure of the largest intersecting families in a wide variety of incidence structures. We say that an incidence structure satisfies the strong EKR property if all intersecting families of maximum size consist of all the blocks through a fixed point.

In this talk I will discuss this problem in ovoidal circle geometries. They arise by taking a quadratic surface Q in PG(3,q) (which is a slight generalisation of a classical polar space) and taking the plane sections with every plane that intersects Q in an oval. I will discuss the proof that the strong EKR property holds in Möbius planes of even order greater than two, and in ovoidal Laguerre planes. As a corollary, the strong EKR property also holds for polynomials of bounded degree over a finite field.

The proof is an illustration of the beautiful marriage of Erdős-Ko-Rado problems and algebraic graph theory.
Period5 Oct 2021
Held atUniversity of Campania Luigi Vanvitelli, Italy