Archive for Mathematical Logic 58 (3-4):469-483 (2019)
AbstractErdős proved that for every infinite \ there is \ with \, 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, \ is a finite set of formulas of T, \, and X is an infinite subset of M. Then there is \ with \ and an equivalence relation E on Y with infinitely many classes, each class infinite, such that Y is \\)-indiscernible. We also consider the definable version of these problems, for example we assume \ is perfect and we find some perfect \ with all distances distinct. Finally we show that Erdős’s theorem requires some use of the axiom of choice.
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Characterizing Model-Theoretic Dividing Lines Via Collapse of Generalized Indiscernibles.Vincent Guingona, Cameron Donnay Hill & Lynn Scow - 2017 - Annals of Pure and Applied Logic 168 (5):1091-1111.
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