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

Abstract

A structure ${\mathcal A} = (A;E_0,E_1, \dots, E_{n-1})$ 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 $X \cap Y$ is finite. A coloring $\chi \colon A \longrightarrow n$ is {\it acceptable} if whenever $X$ is an equivalence class of $E_i$, then $\{x \in X \colon \chi(x) = i\}$ is finite. If $B$ is any set, then the $n$-cube $B^n = (B^n;E_0,E_1, \dots, E_{n-1})$ 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 Sierpiński [17], proved that $\mathbb{R}^n$ has an acceptable coloring iff $2^{\aleph_0} \leq \aleph_{n-2}$. The main result is: if ${\mathcal A}$ is a semialgebraic (i.e., first-order definable in the field of reals) $n$-grid, then the following are equivalent: (1) if ${\mathcal A}$ embeds all finite $n$-cubes, then $2^{\aleph_0} \leq \aleph_{n-2}$; (2) if ${\mathcal A}$ embeds $\mathbb{R}^n$, then $2^{\aleph_0} \leq \aleph_{n-2}$; (3) ${\mathcal A}$ has an acceptable coloring.