Convex decompositions in the plane and continuous pair colorings of the irrationals (Q1852729)

The authors prove that any closed set \(S\) of Euclidean plane which is not a countable union of convex sets satisfies exactly one of the following conditions: \ (1) there exists a perfect nonempty subset \(P\) of \(S\) such that \(|C\cap P|<3\) for every convex subset \(C\) of \(S\), \ (2) there exists a continuous pair coloring \(f: [{\mathcal N}]^2 \to \{0,1\}\) of the space \(\mathcal N\) of irrational numbers so that the covering of \(S\) by convex sets is equivalent to the covering of \(\mathcal N\) by \(f\)-monochromatic sets.