Hilton-Milner results in projective and affine spaces

TL;DR

This paper investigates the structure of the second largest maximal sets of k-spaces in projective and affine spaces that intersect pairwise in at least a t-space, extending known results about the largest configurations.

Contribution

It characterizes the structure of the second largest maximal examples of k-spaces with pairwise t-space intersections in PG(n,q) and AG(n,q).

Findings

Identifies the second largest maximal sets in PG(n,q) and AG(n,q).

Provides structural analysis of these second largest sets.

Extends Hilton-Milner type results to geometric space configurations.

Abstract

In this article, we analyse maximal sets of $k$-spaces, in PG(n,q) and AG(n,q), $n>2k+t+2$, that pairwise meet in at least a $t$-space. It is known that for both PG(n,q) and AG(n,q), the largest example is a $t$-pencil, i.e. the set of all $k$-spaces containing a fixed $t$-space. In this paper, we analyse the structure of the second largest maximal example in both PG(n,q) and AG(n,q).