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Weak Borel chromatic numbers

机译:弱Borel色数

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Given a graph G whose set of vertices is a Polish space X, the weak Borel chromatic number of G is the least size of a family of pairwise disjoint G-independent Borel sets that covers all of X. Here a set of vertices of a graph G is independent if no two vertices in the set are connected by an edge. We show that it is consistent with an arbitrarily large size of the continuum that every closed graph on a Polish space either has a perfect clique or has a weak Borel chromatic number of at most 1. We observe that some weak version of Todorcevic’s Open Coloring Axiom for closed colorings follows from MA. Slightly weaker results hold for Fσ-graphs. In particular, it is consistent with an arbitrarily large size of the continuum that every locally countable Fσ-graph has a Borel chromatic number of at most 1. We refute various reasonable generalizations of these results to hypergraphs.
机译:给定图G的顶点集为波兰空间X,则G的弱Borel色数是覆盖所有X的成对不相交G无关的Borel集的族的最小大小。在这里,图的一组顶点如果集合中没有两个顶点通过边连接,则G是独立的。我们证明与任意大的连续体大小相一致的是,波兰空间上的每个闭合图要么具有完美的团簇,要么具有弱的Borel色数至多1。我们观察到Todorcevic的Open Coloring Axiom的某些弱形式。对于封闭的颜色,请参见MA。 Fσ-图的结果略弱。特别地,每个连续可数的Fσ图具有不超过1的Borel色数,这与连续体的任意大尺寸相一致。我们将这些结果的各种合理概括驳斥为超图。

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