Stability of Erd\H{o}s-Ko-Rado Theorems in Circle Geometries

TL;DR

This paper investigates the stability and structure of largest intersecting families in finite circle geometries, extending known results to M"obius planes of odd order and non-ovoidal geometries, with implications for geometric combinatorics.

Contribution

It characterizes largest intersecting families in M"obius planes of odd order and non-ovoidal geometries, and establishes a stability result for large families in finite circle geometries.

Findings

Largest intersecting families are circles through a fixed point in M"obius planes of order > 3.

Similar results hold for non-ovoidal circle geometries.

Large intersecting families must contain circles through a common point or nucleus.

Abstract

Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in 3-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal circle geometries, except for M\"obius planes of odd order. In this paper we show that also in these M\"obius planes, if the order is greater than 3, the largest intersecting families are the sets of circles through a fixed point. We show the same result in the only known family of finite non-ovoidal circle geometries. Using the same techniques, we show a stability result on large intersecting families in all ovoidal circle geometries. More specifically, we prove that an intersecting family $\mathcal F$ in one of the known finite circle geometries of order $q$, with $|\mathcal F| \geq \frac 1 {\sqrt2} q^2 + 2 \sqrt 2 q + 8$, must consist of circles through a common point, or through a common nucleus in case of a Laguerre plane of even order.