Distinct volume subsets via indiscernibles

Abstract

Erdős proved that for every infinite $X \subseteq \mathbb {R}^d$ there is $Y \subseteq X$ with $|Y|=|X|$, such that all pairs of points from Y have distinct distances, and he gave partial results for general a-ary volume. In this paper, we search for the strongest possible canonization results for a-ary volume, making use of general model-theoretic machinery. The main difficulty is for singular cardinals; to handle this case we prove the following. Suppose T is a stable theory, $\Delta$ is a finite set of formulas of T, $M \models T$, and X is an infinite subset of M. Then there is $Y \subseteq X$ with $|Y| = |X|$ and an equivalence relation E on Y with infinitely many classes, each class infinite, such that Y is $(\Delta , E)$-indiscernible. We also consider the definable version of these problems, for example we assume $X \subseteq \mathbb {R}^d$ is perfect (in the topological sense) and we find some perfect $Y \subseteq X$ with all distances distinct. Finally we show that Erdős’s theorem requires some use of the axiom of choice.