Triangle‐Free Partitions of Euclidean Space

The instance of this theorem when n = 2 was proved by Ceder [1]. Komjath [4] extended Ceder's result to IR, but with the weaker conclusion that no set of the partition contains the vertices of a regular tetrahedron. Subsequently, we extended Komjath's result to arbitrary dimension n in [7], where it was proved that there is a partition of IR" into countably many sets no one of which contains n +1 pairwise equidistant points. The theorem in [7] is actually a very mild generalization of this; the theorem of this paper can be similarly generalized, and we state this generalization in a concluding remark. Erdos (see [5]) has conjectured that there is a partition of IR into countably many sets no one of which contains the vertices of an isosceles triangle. An appropriate version of this conjecture for n = 1 was proved, assuming CH, by Erdos and Kakutani in [3]. It was proved, still assuming CH, by Davies [2] for n = 2 and by Kunen [5] for arbitrary n. It is still open whether or not the Erdos conjecture can be proved without additional set-theoretic hypotheses. For the remainder of this paper we fix an integer n ^ 2. We consider IR" as a vector space over U. For a,beU, we let ||a|| be the Euclidean norm of a, and we let ab be the usual inner product. Three points a,b, CEU" are the vertices of an equilateral triangle if and only if