The coloring of higher dimensional ball packings. 

A ball packing is a collection of balls with disjoint interiors.  The tangency graph of a ball packing takes the balls as vertices and connect two vertices if and only if they are tangent.  [Planar graphs are the tangency graphs of 2-dimensional disk packings](http://en.wikipedia.org/wiki/Circle_packing_theorem).  So the following is a generalization of four-color theorem.

> **Question**: what is the maximum chromatic number $\chi_d$ for the tangency graph of a ball packing in dimension $d$?

I believe that everyone can prove that $\chi_d\le\kappa_d+1$ where $\kappa_d$ is the kissing number.  The question was asked by [Bagchi and Datta (2012)](http://arxiv.org/abs/1202.0153) who gave the trivial lower bound $\chi_d\ge d+2$.

As far as I know, [Maehara (2007)](http://link.springer.com/article/10.1007%2Fs00373-007-0702-7) first attack the problem for dimension $3$.  His construction for lower bound uses Moser spindle.  It generalizes to higher dimensions and gives $\chi_d\ge d+3$.

The problem has also been [asked on MO](http://mathoverflow.net/q/8232/20595). The only new result appears in [an answer of Cantwell](http://mathoverflow.net/a/8488/20595).  He proposed two constructions: One uses binary code and gives quadratic lower bound, but **I think he made some mistakes**.  The other uses halved cubes.  It can be generalized to higher dimensions by a result of [Liniail, Mishulam and Tarsi (1988)](http://www.cs.huji.ac.il/~nati/PAPERS/matroidal_bijections.pdf) and gives $\chi_d\ge d+4$ for infinitely many $d$.

So the current status is: $d+4\le\chi_d\le\kappa_d+1$.  There is a large gap in between.