关于F. Galvin和R. Rado
A note on conjectures of F. Galvin and R. Rado
摘要：
在1968年，高尔文证明了不可数偏序集$P$是可数多个链的并集当且仅当这对每个大小为$\aleph_1$的子集$Q \subseteq P$都成立。在1981年，Rado提出了一个类似的猜想：一个不可数区间图$G$是可数色的当且仅当对每个诱导子图$H \subseteq G$都是可数色的，其大小为$\aleph_1$。Todorcevic证明了Rado猜想相对于超紧基数的存在是相容的，而Galvin猜想的相容性仍然是开放的。在本文中，我们调查和收集的各种结果有关这两个架构。我们还证明了Rado猜想推广到所有弦图类与超紧基数的存在是相对一致的。
英文摘要：
In 1968, Galvin conjectured that an uncountable poset $P$ is the union of countably many chains if and only if this is true for every subposet $Q \subseteq P$ with size $\aleph_1$. In 1981, Rado formulated a similar conjecture that an uncountable interval graph $G$ is countably chromatic if and only if this is true for every induced subgraph $H \subseteq G$ with size $\aleph_1$. Todorcevic has shown that Rado's Conjecture is consistent relative to the existence of a supercompact cardinal, while the consistency of Galvin's Conjecture remains open. In this paper, we survey and collect a variety of results related to these two conjectures. We also show that the extension of Rado's conjecture to the class of all chordal graphs is relatively consistent with the existence of a supercompact cardinal.
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