ML-1996-10: On a Dichotomy related to Colourings of Definable Graphs in Generic Models

ML-1996-10: Kanovei, Vladimir (1996) On a Dichotomy related to Colourings of Definable Graphs in Generic Models. [Report]

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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 R­OD 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 (al­ though 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.

Item Type: Report
Report Nr: ML-1996-10
Series Name: Mathematical Logic and Foundations (ML)
Year: 1996
Date Deposited: 12 Oct 2016 14:40
Last Modified: 12 Oct 2016 14:40
URI: https://eprints.illc.uva.nl/id/eprint/1378

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