Axiom of choice and chromatic number: examples on the plane
Axiom of choice and chromatic number: examples on the plane
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选择公理和色数：平面上的例子
DOI：
10.1016/j.jcta.2004.01.001
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发表时间：
2004
期刊：
影响因子：
--
通讯作者：
S. Shelah
中科院分区：
文献类型：
--
作者：
A. Soifer;S. Shelah
在我们以前的论文（J. Combin。Theory Ser. A 103（2）（2003）387）我们公式化了一个条件色数定理，它描述了一种设置，其中平面的色数根据集合论的公理取两个不同的值。我们还构造了一个例子，一个距离图的真实的线R的色数取决于系统的公理，我们选择的集合论。本文将其推广到平面R2上的距离图G2的构造，从而更接近于平面问题色数的设定。G2的色数在Zermelo-Fraenkel-Choice公理系统中是4，并且在Solovay（Ann.Math.92 Ser.2（1970）1）研究的具有有限选择的相容公理系统中是不可数的（如果它存在）。
In our previous paper (J. Combin. Theory Ser. A 103 (2) (2003) 387) we formulated a conditional chromatic number theorem, which described a setting in which the chromatic number of the plane takes on two different values depending upon the axioms for set theory. We also constructed an example of a distance graph on the real line R whose chromatic number depends upon the system of axioms we choose for set theory. Ideas developed there are extended in the present paper to construct a distance graph G2on the plane R2, thus coming much closer to the setting of the chromatic number of the plane problem. The chromatic number of G2is 4 in the Zermelo–Fraenkel–Choice system of axioms, and is not countable (if it exists) in a consistent system of axioms with limited choice, studied by Solovay (Ann. Math. 92 Ser. 2 (1970) 1).