关于一般模型中可定义图的着色的二分法
On a dichotomy related to colourings of definable graphs in generic models
摘要：
本文证明了Solovay模型中实数上的OD图G满足以下两个条件之一：（I）G允许OD染色;（II）存在G到G的连续同态，其中G 0是某个F_sigma局部可数图，且不是Solovay模型中的R-OD染色.如果图G是局部可数的或非循环的，则（II）可以通过同态是1-1映射的要求来加强，即嵌入。作为第二个主要结果，我们证明了Sigma^1_2图在可构造宇宙L的集类扩张中允许二分法（I）与二分法（II）（尽管现在（I）和（II）可能是一般相容的）。在这种情况下，如果图是局部可数的，则（I）可以被加强为存在可数序数的Δ ^1_3着色。证明是基于由$\od$集生成的拓扑。
英文摘要：
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.
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