Infinite combinatorics and definability
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.
scientific article; zbMATH DE number 4091491
scientific article