Covering points with planes

TL;DR

This paper investigates the maximum size of point sets in vector spaces that are not covered by certain unions of planes, providing bounds and examples, and extending the problem to matroids.

Contribution

It establishes a general upper bound on the size of such point sets and explores cases where the bound does not hold, including for specific algebraic structures and matroids.

Findings

Proved a tight upper bound for point sets covered by unions of planes of fixed dimension.

Constructed examples where the upper bound does not hold in certain algebraic settings.

Extended the analysis to the context of general matroids.

Abstract

Suppose that each proper subset of a set $S$ of points in a vector space is contained in the union of planes of specified dimensions, but $S$ itself is not contained in any such union. How large can $|S|$ be? We prove a general upper bound on $|S|$, which is tight in some cases, for example when all of the planes have the same dimension. We produce an example showing that this upper bound does not hold for point sets whose proper subsets are covered by lines in $(\mathbb{Z}/p^k\mathbb{Z})^2$ with $k\geq 2$, and prove an upper bound in this case. We also investigate the analogous problem for general matroids.