On the upper chromatic number and multiplte blocking sets of PG($n,q$)

TL;DR

This paper studies the upper chromatic number of hypergraphs formed by points and subspaces in projective geometry, establishing bounds and properties related to blocking sets and coloring constraints.

Contribution

It provides new bounds and characterizations for the upper chromatic number in projective geometries, linking it with properties of blocking sets and coloring strategies.

Findings

Sharp bounds for the upper chromatic number in many cases.

Characterization of small t-fold blocking sets in PG(n,p).

Relation between blocking sets and coloring constraints.

Abstract

We investigate the upper chromatic number of the hypergraph formed by the points and the $k$-dimensional subspaces of $\mathrm{PG}(n,q)$; that is, the most number of colors that can be used to color the points so that every $k$-subspace contains at least two points of the same color. Clearly, if one colors the points of a double blocking set with the same color, the rest of the points may get mutually distinct colors. This gives a trivial lower bound, and we prove that it is sharp in many cases. Due to this relation with double blocking sets, we also prove that for $t\leq \frac38p+1$, a small $t$-fold (weighted) $(n-k)$-blocking set of $\mathrm{PG}(n,p)$, $p$ prime, must contain the weighted sum of $t$ not necessarily distinct $(n-k)$-spaces.