Given a point P in the plane, let SP be the set of points whose distance from P is irrational. What is the smallest number of such sets whose union is the entire plane?
Solution
We clearly need at least 3 sets, since given any two points P and Q we can find a point R whose distance from each of P and Q is rational (indeed integral).
However, 3 suffice. Take any two distinct points P and Q. There are at most two points with given rational distances x from P and y from Q, so there are only countably many points X outside SP and SQ. Each of the these points X rules out as the third point countably many points on the line PQ (namely those which are a rational distance from X). So only countably many points are ruled out in total from a line with uncountably many points. Take one of the remaining points.
© John Scholes
jscholes@kalva.demon.co.uk
1 Jan 2001