Vector spaces as unions of proper subspaces

TL;DR

This paper establishes the minimal number of subspaces of fixed codimension required to cover any vector space over any field, connecting to partitioning finite sets into subspaces.

Contribution

It provides a sharp bound for the number of subspaces needed to cover vector spaces, extending understanding of subspace coverings across different fields.

Findings

Derived a precise bound for subspace coverings

Connected covering problems to finite set partitioning

Applicable to vector spaces over any field

Abstract

In this note, we find a sharp bound for the minimal number (or in general, indexing set) of subspaces of a fixed (finite) codimension needed to cover any vector space V over any field. If V is a finite set, this is related to the problem of partitioning V into subspaces.