On a dichotomy related to colourings of definable graphs in generic models

Abstract

We prove that in the Solovay model every OD graph G on reals satisfies one and only one of the following two conditions: (I) G admits an OD colouring by ordinals; (II) there exists a continuous homomorphism of G_0 into G, where G_0 is a certain F_sigma locally countable graph which is not R-OD colourable by ordinals in the Solovay model. If the graph G is locally countable or acyclic then (II) can be strengthened by the requirement that the homomorphism is a 1-1 map, i.e. an embedding. As the second main result we prove that Sigma^1_2 graphs admit the dichotomy (I) vs. (II) in set--generic extensions of the constructible universe L (although now (I) and (II) may be in general compatible). In this case (I) can be strengthened to the existence of a Delta^1_3 colouring by countable ordinals provided the graph is locally countable. The proofs are based on a topology generated by $\od$ sets.