Partial k-Parallelisms in Finite Projective Spaces

TL;DR

This paper investigates the maximum number of pairwise disjoint k-spreads in finite projective spaces, establishing existence results under certain divisibility conditions and extending known parallelism results in projective geometries.

Contribution

It proves the existence of multiple disjoint k-spreads under specific divisibility conditions and extends parallelism results in projective geometries.

Findings

Existence of at least two disjoint k-spreads when k+1 divides n+1 and n>k.

Existence of at least 2^{k+1}-1 pairwise disjoint k-spreads in PG(n,2).

Extension of parallelism results in projective geometries with points removed.

Abstract

In this paper we consider the following question. What is the maximum number of pairwise disjoint $k$-spreads which exist in PG(n,q)? We prove that if k+1 divides n+1 and n>k then there exist at least two disjoint k-spreads in PG(n,q) and there exist at least $2^{k+1}-1$ pairwise disjoint $k$-spreads in PG(n,2). We also extend the known results on parallelism in a projective geometry from which the points of a given subspace were removed.