**** Partial spoiler alert ****

...

"Can we colour the points in R^3 with the colours red, green and blue such that every line parallel to the x-axis contains only finitely many red points, every line parallel to the y-axis contains only finitely many green points, and every line parallel to the z-axis contains only finitely many blue points?"

 Well, I'll have to buy this one since nobody else has volunteered --- just how does the answer to this intriguingly elementary question depend on the continuum hypothesis?

There exists such a colouring if and only if there are no sets of intermediate cardinality between |N| and |R|. In particular, you can derive a contradiction if you assume there's such a colouring of R x V x N (where V is a set of intermediate cardinality), thus there are no colourings of R x R x R. For the converse, it is not too difficult to construct a colouring of O x O x O, where O is the set of all countable ordinals (i.e. ordinals below omega_1). The continuum hypothesis asserts a bijection between O and R, so a solution for O x O x O (which definitely exists) implies an equivalent solution for R x R x R. The problem generalises to k colours in R^k. By the same argument, it's possible to show that there is such a colouring if and only if R is at most the (k-1)th smallest infinite set. I'll give the details of the construction and the reductio ad absurdum if you're interested.

 [I've a feeling that I once knew Imre Leader slightly, a long time ago when we were both postgraduate students; but can no longer recall where or how we met.]

It's a small world. Sincerely, Adam P. Goucher http://cp4space.wordpress.com