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**Moshe Rosenfeld’s odd-distance problem:** Let G be the graph whose vertices are points in the plane and two vertices form an edge if their distance is an odd integer. Is the chromatic number of this graph finite?

Here is a link to a paper on the problem by Hayri Ardal, Jan Manuch, Moshe Rosenfeld, Shaharon Shelah, and Ladislav Stacho.

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Update: Moshe is here, and he just told me that it follows from a theorem of Furstenberg, Katzenelson, and Weiss that if every color class is measurable than infinitely many colors are needed!