Convex decompositions in the plane and continuous pair colorings of the irrationals

We address the structure of nonconvex closed subsets of the Euclidean plane. A closed subset S ⊆ ℝ² which is not presentable as a countable union of convex sets satisfies the following dichotomy: (1) There is a perfect nonempty P ⊆ S so that |C ∩ P| < 3 for every convex C ⊆ S. In this case covering S by convex subsets of S is equivalent to covering P by finite subsets, hence no nontrivial convex covers of S can exist. (2) There exists a continuous pair coloring f: [N]² → (0, 1) of the space N of irrational numbers so that covering S by convex subsets is equivalent to covering N by f-monochromatic sets. In this case it is consistent that S has a convex cover of cardinality strictly smaller than the continuum c in some forcing extension of the universe. We also snow that if f: [N]² → (0, 1) is a continuous coloring of pairs, and no open subset of N is f-monochromatic, then the least number k of f-monochromatic sets required to cover N satisfies k⁺ ≥ c. Consequently, a closed subset of ℝ² that cannot be covered by countably many convex subsets, cannot be covered by any number of convex subsets other than the continuum or the immediate predecessor of the continuum. The analogous fact is false for closed subsets of ℝ³.