Continuous 2-colorings and topological dynamics
Volume 586 / 2023
Abstract
We first consider the class $\mathfrak K$ 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 $\mathfrak K$ made up of countable graphs, comparing them with the quasi-order $\preceq^i_c$ 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 $\mathfrak K$, and provide infinite descending chains in $\mathfrak K$. Our method implies that the equivalence relation of flip conjugacy of minimal homeomorphisms of $2^\omega$ is Borel reducible to the equivalence relation associated with $\preceq^i_c$. 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 $\preceq^i_c$-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.
