Published online by Cambridge University Press: 14 June 2011
Given a set D of positive real numbers, let Xn(D) denote the graph with ℝn as the vertex set such that two points are joined if their distance is in D. Bukh conjectured in [Measurable sets with excluded distances. Geom. Funct. Anal.18 (2008), 668–697] that if D is algebraically independent, then Chr(Xn(D)), the chromatic number of Xn(D), is finite. Here we prove that Chr(Xn(D)) is countable and that, if n=2 , even the coloring number is countable. Furthermore, we prove that Chr (Y ) is countable, where Y is the following graph on ℂn: let 𝔽 be a countable subfield of ℂ and let D⊆ℂ be algebraically independent over 𝔽; join a,b∈ℂn if there is some p(x,y)∈𝔽[x,y] such that p(x,x) is identically zero and p(a,b)≠0 is algebraic over some d∈𝔽∪D.