TITLE: Zorn's Lemma and plane geometry
QUESTION [5 upvotes]: Given a graph $G$ and a number $n$, Zorn's Lemma immediately implies the existence of a
maximal partial coloring of $G$.  Equivalently, one may assign $n+1$ colors to the nodes of $G$ such that nodes with the first $n$ colors never lie adjacent, but every node of the last color lies adjacent to at least one node receiving each of the other colors.
In particular, the observation above applies to geometrically determined graphs and even more specifically to graphs on, say, ${\Bbb R}^2$, where one declares two points $p_1$ and $p_2$  adjacent provided that their distance $d(p_1,p_2)$ lies in $D$, some prescribed set of distances.  
I would like to know what values of $n$ and $D$ make this existence result  not a theorem of ZF.  If that asks too much, any example of such an $n$ and $D$ would satisfy me, even better, the simplest possible example (say with $|D|$ as small as possible, and $n$ as small as possible given that).

REPLY [3 votes]: There is some recent material by Michael Payne that is also relevant here:  
He considers the following variant of the Shelah-Soifer graph:  Two points in the plane form
an edge if the distance is 1, the $x$-coordinates have rational difference, and the $y$-coordinates have rational difference.  
The chromatic number of this graph is 2 (in ZFC).  The measurable chromatic number (preimage of colors Lebesgue measurable) is at least 5.  If all subsets of the plane are measurable (e.g., in Solovay's model), the chromatic number of this graph is at least 5.  
Unfortunately I cannot give you a link to the paper at the very moment. 
I believe the general problem that you are asking about is open.