For a set of points in $\mathbb{R}^d$ with minimum distance $a$, the minimum-distance graph connect two points iff they are at distance $a$. We can also view it as the tangency graph for a set of unit balls, so the upper bound for the chromatic number is obviously the one-side kissing number.

It is known that for large dimensions, the chromatic number for *unit-distance graphs* has an exponential lower bound (Frankl and Wilson), and the chromatic number for *largest-distance graphs* has a sub-exponential lower bound (Kahn and Kalai).

What is the best known bound for the chromatic number for minimum-distance graphs? Is there already a (sub-)exponential lower bound or a polynomial upper bound?