Submission date: 17 July 2018

Abstract:

Erdös proved that for every infinite X ⊆ ℝ^d there is Y ⊆ 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 theb following. Suppose T is a stable theory, Δ is a finite set of formulas of T, M ⊨ T, and X is an infinite subset of M. Then there is Y ⊆ X with |Y| = |X| and an equivalence relation E on Y with infinitely many classes, each class infinite, such that Y is (Δ, E)-indiscernible. We also consider the definable version of these problems, for example we assume X ⊆ ℝ^d is perfect (in the topological sense) and we find some perfect Y ⊆ X with all distances distinct. Finally we show that Erdös's theorem requires some use of the axiom of choice.

Mathematics Subject Classification: 03E75, 03C98

Keywords and phrases:

Full text arXiv 1807.06654: pdf, ps.