Consider an infinite integer lattice, or an infinite hexagonal lattice with unit length edges. Provided a set of $k$ possible vertex colors, is there a known largest possible minimum spacing that can be achieved between vertices of the same color? What about for three-dimensional integer lattice?
Turning this question around, you can instead ask how many colours you need to ensure that no two points at distance at most $d$ receive the same colour. That is, add edges to your graph between all pairs of vertices at distance at most $d$: what is the chromatic number of this new graph?
If we join points at distance exactly $d$ instead of at most $d$, then this question has received quite a lot of attention. It is a famous open problem to determine the chromatic number of the plane, the chromatic number of $\mathbb R^2$ with edges between points at Euclidean distance 1. There are easy constructions to show that $4 \leq \chi(\mathbb R^2) \leq 7$, but nothing better is known.
The construction for the lower bound generalises to give $\chi(\mathbb R^n) \geq n+2$ for every $n$. The first exponential lower bound was proved by Frankl and Wilson. A set system can be embedded as a collection of 0-1 vectors, so that distances correspond to intersection sizes. Frankl and Wilson spotted that they could apply a result of theirs on forbidden intersection sizes to show that independent sets in their collection of vectors could not be too large, which was enough to prove that $\chi(\mathbb R^n) \geq (1.05+o(1))^n$.