Axiom of choice and chromatic number of Rn
Axiom of choice and chromatic number of Rn
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Rn 的选择公理和色数
DOI：
10.1016/j.jcta.2004.10.004
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
A. Soifer
中科院分区：
文献类型：
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作者：
A. Soifer
在以前的论文中(J·康宾理论系列。A 103(2003)387)和(J.Combin.理论系列。A 105(2004)359)Saharon Shelah和我建立了一个条件色数定理，该定理描述了平面的色数根据集合论的公理而具有两个不同的值的设置。我们还构造了实直线R上的距离图和实平面R2上的差图的例子，它们的色数取决于我们为集合论选择的公理系统。本文将本文发展的思想推广到构造实空间Rn上的差图，其色数在Zermelo-Fraenkel-Choice公理系统中为正整数，在有限选择的相容公理系统中不可数(如果存在)，由Solovay(Ann)研究。数学课。先生。2(1970)1)。这些例子说明了组合结果在很大程度上依赖于基本的集合论，有助于理解n空间的色数问题的潜在复杂性，并表明n空间的色数可能取决于为集合论选择的公理系统。
In previous papers (J. Combin Theory Ser. A 103 (2003) 387) and (J. Combin. Theory Ser. A 105 (2004) 359) Saharon Shelah and I 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 examples of a distance graph on the real line R and difference graphs on the real plane R2whose chromatic numbers depend upon the system of axioms we choose for set theory. Ideas developed there are extended in the present paper to construct difference graphs on the real space Rn, whose chromatic number is a positive integer 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. Ser. 2 (1970) 1). These examples illuminate how heavily combinatorial results can depend upon the underlying set theory, help appreciate the potential complexity of the chromatic number of n-space problem, and suggest that the chromatic number of n-space may depend upon the system of axioms chosen for set theory.