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arXiv 2012.146382020-12-29math.LOmath.CO

Maximal discrete sets

David Schrittesser

备注：: 21 pages. Contains some corrections in comparison to the published version
发表信息：: Set Theory and Infinity, RIMS Kôkyûroku No.2164, 2019

英文摘要：

We survey results regarding the definability and size of maximal discrete sets in analytic hypergraphs. Our main examples include maximal almost disjoint (or mad) families, $\mathcal I$-mad families, maximal eventually different families, and maximal cofinitary groups. We discuss the non-increasing sequence of cardinal characteristics $\mathfrak a_ξ$, for $ξ<ω_1$ as well as the notions of spectra of characteristics and optimal projective witnesses. We give an account of Zhang's forcing to add generic cofinitary permutations, and of a version of this forcing with built-in coding.

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