The arc chromatic number for Galois projective planes, affine planes and Euclidean grids

Abstract

In 1986, Csima and Füredi determined the minimum number of arcs required to partition the points of Galois projective planes $PG(2,q)$ and affine planes $AG(2,q)$. We revisit these foundational results and extend them in several directions. First, we present alternative proofs for some of their results, providing a more streamlined approach. By leveraging this new perspective, we determine the exact value for a fractional version of the problem in both geometric settings. We also establish a computational framework that yields new constructions and explores the structural properties of the space of colorings. Finally, we apply these findings to Euclidean grids, improving upon a 2004 construction by Wood. We prove that a partition into $(1+\epsilon )n$ sets in general position exists for any $\epsilon >0$ and sufficiently large $n$. Additionally, we provide exact minimal partitions for small Euclidean grids.

22 pages, 3 figures. Revised historical positioning, including the earlier work of Csima-Füredi and Wood. New results for the fractional arc chromatic number and non-uniqueness of optimal colorings. The computational and Euclidean-grid sections have also been expanded, including recent results on $G_9$ and $G_{10}$