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The answer is 5, 6 or 7. The lower bound of 5 came in 2018 from Aubrey de Grey, an amateur, who found a unit-distance graph needing five colors. A Polymath project then shrank his graph from 1,581 vertices to about 500. The answer may depend on the axioms of set theory if the coloring is not required to be measurable.
Posed 1950, Edward Nelson.