Distinct Volume Subsets via Indiscernibles

Abstract

Erd\"{o}s 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 Erd\"{o}s's theorem requires some use of the axiom of choice.