It is a famous open problem to determine the chromatic number of the graph $G=G(\mathbb{E}^2,\{1\})$ whose vertices are the point in the plane and two vertices are linked by an edge when they are at Euclidean distance $1$ one from the other. This number is known to be between $4$ and $7$.

It is a theorem of De Bruijn and Erdös that the axiom of choice implies that the chromatic number of a graph is the maximal chromatic number of its finite subgraphs. As the axiom of choice does matter in certain such coloring problem, it is worth considering the following as a slight variant of the above question: what is the maximal chromatic number of a finite subgraph of $G$?

Since this question is obviously too difficult, let us ask about a stronger statistic: what is the maximal minimum degree of finite subgraphs of $G$?
Recall that the chromatic number of a graph is at most its *degeneracy* plus one, where degeneracy is the least number $k$ such that inductively pruning the vertices of degree at most $k$ empties the graph.

I expect this question to be open, and my real question is (at last) the following:

What is the maximal minimum degree of

knownfinite subgraphs of $G$?

I ask it because most unit-distance graphs I saw have a vertex of degree $3$. If this where true for all finite subgraphs of $G$, then it would have chromatic number $4$; but there are $4$-regular finite subgraphs of $G$. Does anyone knows how to achieve $5$?