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A generalization of Sierpiński's paradoxical decompositions: Coloring semialgebraic grids

Published online by Cambridge University Press:  12 March 2014

James H. Schmerl
James H. Schmerl*
Affiliation:
Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, E-mail: james.schmerl@uconn.edu
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Abstract

A structure is an n-grid if each Ei, is an equivalence relation on A and whenever X and Y are equivalence classes of, respectively, distinct Ei, and Ej, then X ∩ Y is finite. A coloring χ: A → n is acceptable if whenever X is an equivalence class of Ei, then {x ∈ X: χ(x) = i} is finite. If B is any set, then the n-cube Bn = (Bn; E0, …, En−1) is considered as an n-grid, where the equivalence classes of Ei are the lines parallel to the i-th coordinate axis. Kuratowski [9], generalizing the n = 3 case proved by Sierpihski [17], proved that ℝn has an acceptable coloring iff 2ℵ0 ≤ ℵn−2. The main result is: if is a semialgebraic (i.e., first-order definable in the field of reals) n-grid, then the following are equivalent: (1) if embeds all finite n-cubes, then 2ℵ0 ≤ ℵn−2: (2) if embeds ℝn, then 2ℵ0 ≤ ℵn−2; (3) has an acceptable coloring.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 4 , December 2012 , pp. 1165 - 1183
Copyright
Copyright © Association for Symbolic Logic 2012

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