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

  • James H. Schmerl (a1)

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 XY is finite. A coloring χ: An is acceptable if whenever X is an equivalence class of Ei, then {xX: 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 20 ≤ 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 20 ≤ ℵn−2: (2) if embeds ℝn, then 20 ≤ ℵn−2; (3) has an acceptable coloring.

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

  • James H. Schmerl (a1)

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