Infinite combinatorics and definability
type: Item
instance of: scholarly article
title: Infinite combinatorics and definability
zbMATH Open document ID: 0667.03037
DOI: 10.1016/0168-0072(89)90013-4
author: Arnold W. Miller
published in: Annals of Pure and Applied Logic
publication date: 1989-01-01T00:00:00Z
review text: It is proved that if \(B\subseteq 2^{\omega}\times [\omega]^{\omega}\) is Borel, then there exist a perfect \(C\subseteq 2^{\omega}\) and an infinite \(X\subseteq \omega\) such that \(C\times [X]^{\omega}\) is either contained in or disjoint from B. This can be regarded as a parametrized version of the Galvin-Prikry theorem. Another result is that it is consistent that \(c=\omega_ 3\) and if \(\omega_ 2\) many Borel sets cover R, then some \(\omega_ 1\) of them still cover. Aparametrized version of the \(\Delta\)-system lemma is given, it is also shown that no such version is true for some theorems on strongly disjoint systems. Concerning an old result of Mazurkiewicz it is proved that under \(V=L\) there is a \(\Pi^ 1_ 1\) subset of the plane hitting every line in exactly two points. Similarly, under \(V=L\) there is a \(\Pi_ 1^ 1\) Hamel basis. Sierpinski's theorem stating that Hamel bases cannot be \(\Sigma^ 1_ 1\) is reproved. Similar results are proved on maximal almost disjoint families. Most proofs are via forcing techniques.
reviewed by: Péter Komjáth
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 03E05
Mathematics Subject Classification ID: 03E45
Mathematics Subject Classification ID: 03E35
zbMATH DE Number: 4091491
zbMATH Keywords: Galvin-Prikry theorem
zbMATH Keywords: Borel sets
zbMATH Keywords: \(\Delta\)-system lemma
zbMATH Keywords: Hamel basis
zbMATH Keywords: maximal almost disjoint families
zbMATH Keywords: forcing
MaRDI profile type: Publication
full work available at URL: 0168-0072(89)90013-4
OpenAlex ID: W2013432018
cites work: Iterated perfect-set forcing
cites work: A Partition Theorem For Perfect Sets
cites work: A new proof that analytic sets are Ramsey
cites work: Q3789551
cites work: Some Remarks on Set Theory
cites work: Some additive properties of sets of real numbers
cites work: On a Problem of Erdos on Sequences and Measurable Sets
cites work: Borel sets and Ramsey's theorem
cites work: Q3270120
cites work: Q3903002
cites work: Borel Orderings
cites work: Q3852172
cites work: Measure and other properties of a Hamel basis
cites work: Large Sets not Containing Images of a Given Sequence
cites work: Families close to disjoint ones
cites work: On the limit superior of analytic sets
cites work: Set theory. An introduction to independence proofs
cites work: A Problem of Incidence
cites work: Products of Infinitely Many Perfect Trees
cites work: Q3967533
cites work: Happy families
cites work: Q3715868
cites work: Q3946367
cites work: Some Results Connected with a Problem of Erdos. II
cites work: Descriptive set theory
cites work: Independent sets in topological algebras
cites work: Algebraic independence and measure
cites work: Q3709279
cites work: On co-\(\kappa\)-Souslin relations
cites work: Can you take Solovay's inaccessible away?
cites work: Every analytic set is Ramsey
cites work: A model of set-theory in which every set of reals is Lebesgue measurable
cites work: Compacts de fonctions mesurables et filtres non mesurables
description: scientific article; zbMATH DE number 4091491