Erd\H{o}s-Ko-Rado theorems for ovoidal circle geometries and polynomials over finite fields

TL;DR

This paper extends Erd ext{"o}s-Ko-Rado theorems to ovoidal circle geometries and polynomial families over finite fields, identifying the structure of largest intersecting families in these settings.

Contribution

It establishes the structure of maximum intersecting families of circles and polynomials in various ovoidal geometries and finite fields, including new results for Minkowski planes.

Findings

Largest intersecting circle families pass through a fixed point in certain geometries.

Maximum polynomial families with shared values are characterized by fixed input-output pairs.

Results include new bounds and structures for intersecting families in ovoidal Minkowski planes.

Abstract

In this paper we investigate Erd\H{o}s-Ko-Rado theorems in ovoidal circle geometries. We prove that in M\"obius planes of even order greater than 2, and ovoidal Laguerre planes of odd order, the largest families of circles which pairwise intersect in at least one point, consist of all circles through a fixed point. In ovoidal Laguerre planes of even order, a similar result holds, but there is one other type of largest family of pairwise intersecting circles. As a corollary, we prove that the largest families of polynomials over $\mathbb F_q$ of degree at most $k$, with $2 \leq k < q$, which pairwise take the same value on at least one point, consist of all polynomials $f$ of degree at most $k$ such that $f(x) = y$ for some fixed $x$ and $y$ in $\mathbb F_q$. We also discuss this problem for ovoidal Minkowski planes, and we investigate the largest families of circles pairwise intersecting in two points in circle geometries.