The Ramsey property implies no mad families
The Ramsey property implies no mad families
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拉姆齐财产意味着没有疯狂的家庭
DOI：
10.1073/pnas.1906183116
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
Asger Törnquist
中科院分区：
文献类型：
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作者：
David Schrittesser;Asger Törnquist
现代数学中的某些无限组合结构，称为疯狂的家庭，已知只存在由于间接的，非建设性的方法产生的一个基本原则的数学，与许多矛盾的后果，称为公理的选择。本文表明，如果我们取代公理的选择与普遍的组合规律性的自然假设，一个原则被称为拉姆齐财产的所有集，那么没有无限疯狂的家庭可以存在。这解决了一个自20世纪60年代末以来一直在数学中公开的问题。我们证明了，如果N的无限子集的所有集合具有Ramsey性质，则不存在无限极大几乎不相交（疯狂）的家庭。在Zermelo-Fraenkel集合论中，仅用弱选择原则证明了该蕴涵。这为一个长期存在的问题提供了一个积极的解决方案，这个问题可以追溯到Mathias [A]。R. D. Mathias，Ann. Math. Logic 12，59-111（1977）]。这个证明利用了一个在遍历理论、拓扑动力学和不变描述集理论中有其自然根源的想法：我们使用与一个所谓的疯狂家族相关的某个函数在等价关系E0下是不变的，因此在一个“大”集合上是常数。此外，我们宣布了一些额外的结果，疯狂的家庭相对于更复杂的Borel理想。
Significance Certain infinite combinatorial structures in modern mathematics, called mad families, are known to exist only due to indirect, nonconstructive methods arising from a fundamental principle of mathematics, with many paradoxical consequences, called the axiom of choice. This paper shows that if we replace the axiom of choice with a natural assumption of universal combinatorial regularity, a principle known as the Ramsey property for all sets, then no infinite mad families can exist. This solves a problem that has been open in mathematics since the late 1960s. We show that if all collections of infinite subsets of N have the Ramsey property, then there are no infinite maximal almost disjoint (mad) families. The implication is proved in Zermelo–Fraenkel set theory with only weak choice principles. This gives a positive solution to a long-standing problem that goes back to Mathias [A. R. D. Mathias, Ann. Math. Logic 12, 59–111 (1977)]. The proof exploits an idea which has its natural roots in ergodic theory, topological dynamics, and invariant descriptive set theory: We use that a certain function associated to a purported mad family is invariant under the equivalence relation E0 and thus is constant on a “large” set. Furthermore, we announce a number of additional results about mad families relative to more complicated Borel ideals.