Distinct Volume Subsets via Indiscernibles

TL;DR

This paper explores the structure of large subsets with distinct volume properties in Euclidean spaces using model theory, extending Erdős's results and addressing cases involving singular cardinals and definability.

Contribution

It introduces new canonization results for $a$-ary volume using model-theoretic methods, especially for singular cardinals, and examines definable versions and the necessity of the axiom of choice.

Findings

Existence of large indiscernible subsets with distinct volume properties.

Extension of Erdős's theorem to singular cardinals using stability theory.

Demonstration that Erdős's theorem relies on the axiom of choice.

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.