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    $\begingroup$ Is it known whether this is the same as the chromatic number of maps whose regions are convex subsets of $\mathbb{R}^d$? $\endgroup$ Commented Dec 8, 2014 at 1:05
  • $\begingroup$ @NoahS, This could be arbitrarily high, isn't it? $\endgroup$ Commented Dec 8, 2014 at 8:09
  • $\begingroup$ @NoahS Yes, it's arbitrarily high. See arxiv.org/abs/math/0106095 . So it is known that they are not the same. $\endgroup$ Commented Dec 8, 2014 at 8:17