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

Abstract

We establish that the minimum number of arcs required to partition the Galois projective plane $\text{PG}(2,q)$ is $q+1$. Furthermore, we determine the exact value for a fractional variant of this problem. We extend our analysis to affine planes $\text{AG}(2,q)$, proving that they can be partitioned into $q$ arcs. In particular, we show that this partition is tight when $q$ is an odd prime power, and that a $(q-1)$-partition is attainable for $q=2^k$ with $k \in \{1,2,3\}$. For $q=2^k$ with $k \geq 4$, we provide bounds between two possible values. Finally, we apply these results to Euclidean grids, demonstrating that a partition into $(1+\epsilon)n$ sets in general position exists for any $\epsilon > 0$ and sufficiently large $n$. We also present exact minimal partitions for small Euclidean grids.