The recent striking progress on the chromatic number of the plane by de Grey arises from the interesting fact that certain Cayley graphs have large chromatic number; namely, the graph whose vertices are the ring of integers of a certain number field K endowed with a complex absolute value ||, and in which x and y are adjacent if and only if |x-y| = 1.
This made me realize I know very little about the chromatic numbers of infinite Cayley graphs, and too my surprise, I wasn't able to find much in the literature!
So here's a sample question. Let A be a finitely generated free abelian group, let S be a finite subset of A closed under negation, and let G be the Cayley graph whose vertices are the elements of A and where a,b are adjacent if and only if |a-b| lies in S.
For every integer N, G has a quotient G/N, a finite Cayley graph whose vertices are A/NA and whose edges are given by the images of S in A/NA. (Maybe better to take N large enough so that no element of S lies in NA, so G/N has no loops.)
Evidently a coloring of G/N pulls back to G, so we get an inequality of chromatic numbers
$\chi(G) \leq \chi(G/N)$.
My question is: is it always the case that
$\chi(G) = \inf_N \chi(G/N)$?
In other words: if G has a k-coloring, does it necessarily have a periodic k-coloring?
(You might think of $\inf_N \chi(G/N)$ as the chromatic number of a profinite Cayley graph.)
I don't even know how to do this for A=Z!