2018/07/17 by William Gasarch, Gasarch, William, Douglas Ulrich +1
Mathematics · #03C98 #03E75 #FOS: Mathematics #Logic (math.LO) #math.LO #msc:03C98 #msc:03E75
paper · pdf · doi:10.48550/arxiv.1807.06654
15 pages
arxiv created 2018/07/17 · arxiv updated 2018/07/19
Erdös proved that for every infinite X ⊆ ℝd 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 the following. Suppose T is a stable theory, Δ is a finite set of formulas of T, M \models 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 ⊆ ℝd 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.