Cross-Intersecting Erd\H{o}s-Ko-Rado Sets in Finite Classical Polar Spaces

TL;DR

This paper investigates cross-intersecting Erd ext{"o}s-Ko-Rado sets in finite classical polar spaces, establishing upper bounds on their sizes and classifying maximum configurations, except for Hermitian cases.

Contribution

It provides new upper bounds on the product of sizes of cross-intersecting sets and classifies maximum configurations in most finite polar spaces.

Findings

Upper bounds on |Y| * |Z| for cross-intersecting sets

Classification of maximum size sets in all but Hermitian polar spaces

Extension of Erd ext{"o}s-Ko-Rado theory to polar spaces

Abstract

A cross-intersecting Erd\H{o}s-Ko-Rado set of generators of a finite classical polar space is a pair $(Y, Z)$ of sets of generators such that all $y \in Y$ and $z \in Z$ intersect in at least a point. We provide upper bounds on $|Y| \cdot |Z|$ and classify the cross-intersecting Erd\H{o}s-Ko-Rado sets of maximum size with respect to $|Y| \cdot |Z|$ for all polar spaces except Hermitian polar spaces in odd projective dimension.