Two dichotomy theorems on colourability of non-analytic graphs

We prove: Theorem 1. Let κ be an uncountable cardinal. Every κ-Suslin graph G on reals satisfies one of the following two requirements: (I) G admits a (κ+ 1)-Borel colouring by ordinals below κ; (II) there exists a continuous homomorphism (in some cases an embedding) of a certain locally countable Borel graph G0 into G. Theorem 2. In the Solovay model , every OD graph G on reals satisfies one of the following two requirements: (I) G admits an OD colouring by countable ordinals; (II) as above. Introduction. A new direction in the theory of graphs was discovered by Kechris, Solecki, and Todorčević [9]. They demonstrated that graphs on reals change their behaviour in the case when a definable colouring is required. For instance, there exists a Borel locally countable and acyclic (hence colourable by only two colours) graph G0 on reals (or rather a family of graphs generated by a common method) which does not admit a countable Borel colouring. To get such a graph, let us fix once and for all a binary tuple sm ∈ 2 for each m ∈ ω, so that for any s ∈ 2 we have s ⊆ sm for some m (the density). We set, for x, y ∈ 2, x G0 y iff there exists m satisfying x1m = y1m = sm, x(m) = 1− y(m), and x(k) = y(k) for all k > m. Thus G0 is a Borel (more exactly, Fσ) locally countable acyclic (1) graph on the Cantor space D = 2. An argument of Kechris et al. [9] shows that the density property implies countable Borel uncolourability of G0. 1991 Mathematics Subject Classification: 03E15, 03E35, 04A15. Partially supported by DFG, NWO, and the Carlsberg foundation. (1) A (symmetric irreflexive) graph G is acyclic iff there does not exist a path of the form x0 G x1 G x2 G . . .G xn where n ≥ 3, x0 = xn, and xi 6= xj except for i = 0, j = n. A graph G is locally countable iff every vertex has at most countably many G-neighbours.