The existence of a 4-chromatic unit distance graph (e.g., the Moser spindle) establishes a lower bound of 4 for the chromatic number of the plane (see the Nelson-Hadwiger problem).

Obviously, it would be nice to have an example of a *5-chromatic* unit distance graph. To the best of my knowledge, the existence of such a graph is open. Has there been any (documented) attempt to find such a graph through a computer search? For instance, has every $n$-vertex possibility been checked up to some $n$?

almost-unit distance graph. You can indeed ask for the chromatic number $\chi_\epsilon$ of the graph whose vertices are the points of the plane, and an edge joins two vertices if their distance is between $1$ and $1+\varepsilon$. Obviously $\chi_\varepsilon$ decreases when $\varepsilon\to 0$, but the limit is not known. It is not known if the limit is the chromatic number of the plane either. $\endgroup$