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

A structure is an n-grid if each E i , is an equivalence relation on A and whenever X and Y are equivalence classes of, respectively, distinct E i , and E j , then X ∩ Y is finite. A coloring χ : A → n is acceptable if whenever X is an equivalence class of E i , then { x ∈ X : χ ( x ) = i } is finite. If B is any set, then the n -cube B n = ( B n ; E 0 , …, E n −1 ) is considered as an n -grid, where the equivalence classes of E i are the lines parallel to the i -th coordinate axis. Kuratowski [9], generalizing the n = 3 case proved by Sierpihski [17], proved that ℝ n has an acceptable coloring iff 2 ℵ 0 ≤ ℵ n −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 2 ℵ 0 ≤ ℵ n −2 : (2) if embeds ℝ n , then 2 ℵ 0 ≤ ℵ n −2 ; (3) has an acceptable coloring.