This is the well-known unit distance graph problem. If we call $U=U(\mathbb R^2)$ as the unit distance graph on the plane; that is, the vertices are the points on the plane and the edges are the pairs at distance one from each other.
It is well-known that $$4 \leq \chi(U(\mathbb R^2))\leq 7.$$
The lower bound is found by drawing a finite unit distance subgraph of $U$ which has chromatic number 4 while the upper bound is found by coloring the plane with 7 colors after dividing it into hexagons of a fixed,small diameter.
Recently, I came across the study of the chromatic index of a supergraph of $U$ called odd-distance graphs. Moreover, I think that this problem is equivalent to finding a measure of some sort but that is all I remember.
We once tried to use Hammel Basis of the plane to come up with a proof which at least improves the above bounds but it does not seem to work(well, we were able to prove that $\mathbb Z$ is an integral domain... funny...).