We investigate infinite sets that witness the failure of certain Ramsey-theoretic statements, such as Ramsey's or (appropriately phrased) Hindman's theorem; such sets may exist if one does not assume the Axiom of Choice. We obtain very precise information as to where such sets are located within the hierarchy of infinite Dedekind-finite sets.

中文翻译：

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集合论中的拉姆齐理论陈述引起的有限性类，无选择

我们研究了无穷集，这些无穷集见证了某些Ramsey理论上的陈述的失败，例如Ramsey的定理或Hindman定理（用适当的措词）；如果不假设选择公理，则这些集合可能存在。我们获得了关于此类集合在无限Dedekind有限集合的层次结构中的位置的非常精确的信息。