Partitioning the plane into denumerably many sets without repeated distances

Abstract

Ceder(1) proved (assuming the axiom of choice, as we do throughout this paper) that the Euclidean plane can be partitioned into ℵ0 sets none of which contains an equilateral triangle; indeed he proved that given any denumerable set of triangles, the plane can be partitioned into ℵ0 sets, none containing a triangle similar to one of the given triangles. Erdős and Kakutani(2) proved that the continuum hypothesis implies that the real line can be partitioned into ℵ0, rationally independent sete, and that the existence of such a partition implies the continuum hypothesis. Erdős asked (private communication) whether there is a partition of the plane into ℵ0 sets not containing an isosceles triangle, or more generally in which any four points determine six different distances. Assuming the continuum hypothesis, it will be shown here (Theorem 1) that a, partition of the latter kind does exist. (I communicated this result to Erdős and others some years ago, but subsequently noticed that the argument was incomplete.) Conversely (Theorem 2) the existence of such a partitition, even for the line, implies the continuum hypothesis. This strengthening of the converse half of the Erdős-Kakutani theorem is proved by what is essentially their method (actually in a rather simplified form).