TITLE: Can Sierpinski's anisotropic bicolouring of the plane, assuming the continuum hypothesis (CH), be extended to three dimensions?
QUESTION [5 upvotes]: Sierpinski showed that, on the assumption of CH (in fact, equivalently to it), each point in the plane can be coloured (say) black or white so that every section of the plane parallel to the $x$ axis is "almost" white $-$ in the sense that all but countably many points of it are white $-$ while every section parallel to the $y$ axis is almost black. Equivalently, given CH, every line through the origin can be almost white, while every circle centred on the origin is almost black. Is there a corresponding result for three-dimensional space? For example, assuming CH, can we bicolour space so that every plane through the origin is almost white, while every sphere centred at the origin is almost black?

REPLY [6 votes]: Sierpinski himself proved the following striking version of his theorem for the plane:
CH is equivalent to the statement that $\mathbb R^3$ can be written as $A_x\cup A_y\cup A_z$ where each $A_u$,
$u\in\{x,y,z\}$, has a finite intersection with every line parallel to the $u$-axis.
(See http://www.math.wisc.edu/~miller/old/m873-05/setplane.ps, respectively [Simms, John C.; Sierpinski’s theorem. Simon Stevin 65 (1991), no. 1-2, 69–163].
I believe this theorem is proved in Sierpisnki's 1951 paper in Fund. Math. 38, pp. 1--13, but I would have to actually walk to the library to verify this (and decipher enough of the french in the article).)
There is actually a planar version of this result of Sierpinski, due to Komjath.
A set $A\subseteq\mathbb R^2$ is a cloud if there is $p\in\mathbb R^2$ such that every line through $p$ meets $A$ in only finitely many points.
CH is equivalent to $\mathbb R^2$ being the union of 3 clouds.
In both cases the striking fact is that we go down from countable to finite, which is somehow less arbitrary, in the light of Joel's comment that the previous constructions can be done in ZFC if countable is replaced by $\lneq 2^{\aleph_0}$.