Basis problems in combinatorial set theory
The author discusses results and conjectures concerning some basis problems in set theory. A basis for a class \(S\) of structures is a collection \(S_0\) of critical members of \(S\) such that any given structure from \(S\) can be related via some kind of connecting map to one member from \(S_0\). For instance, the author showed that under PFA, the five sets 1, \(\omega\), \(\omega_1\), \(\omega\times \omega_1\) and \(\text{Fin}_{\omega_1}\) (the collection of all finite subsets of \(\omega_1\) ordered by inclusion) form a basis for the class of all directed sets on \(\omega_1\) in the sense that any such set is Tukey equivalent to one of the five. The author inspects basis problems in a variety of areas, considering successively binary relations on \(\omega_1\), transitive relations on \(\omega_1\), uncountable linear orderings and uncountable regular spaces. He also discusses related Ramsey-theoretic principles such as the Open Coloring Axiom or special Aronszajn trees.
scientific article; zbMATH DE number 1184262