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?



Can Sierpinski's anisotropic bicolouring of the plane, assuming the continuum hypothesis (CH), be extended to three dimensions?
The answer to the last sentence in the question is no, for a silly reason, namely that a plane through the origin and a sphere about the origin have an uncountable intersection, in which only countably many points could be white and countably many black. The obvious way to evade that silliness is to replace planes and spheres with two families of sets, say $X$ and $Y$, such that the intersection of any set in $X$ and any set in $Y$ is countable. If you assume that, and if each of the families $X$ and $Y$ contains only $\aleph\_1$ sets, then Sierpinski's construction will work. 


