Continuous 2-colorings and topological dynamics

We first consider the class K of graphs on a zero-dimensional metrizable compact space with continuous chromatic number at least three. We provide a concrete basis of size continuum for K made up of countable graphs, comparing them with the quasi-order associated with injective continuous homomorphisms. We prove that the size of such a basis is sharp, using odometers. However, using odometers again, we prove that there is no antichain basis in K, and provide infinite descending chains in K. Our method implies that the equivalence relation of flip conjugacy of minimal homeomorphisms of the Cantor space is Borel reducible to the equivalence relation associated with our quasi-order. We also prove that there is no antichain basis in the class of graphs on a zero-dimensional Polish space with continuous chromatic number at least three. We study the graphs induced by a continuous function, and show that any basis for the class of graphs induced by a homeomorphism of a zero-dimensional metrizable compact space with continuous chromatic number at least three must have size continuum, using odometers or subshifts.