Todorcevic, Stevo Basis problems in combinatorial set theory. (English) Zbl 0918.03025 Doc. Math., Extra Vol. ICM Berlin 1998, vol. II, 43-52 (1998). 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. Reviewer: P.Matet (Caen) Cited in 10 Documents MSC: 03E05 Other combinatorial set theory 03E35 Consistency and independence results 03-02 Research exposition (monographs, survey articles) pertaining to mathematical logic and foundations 03E65 Other set-theoretic hypotheses and axioms 54E45 Compact (locally compact) metric spaces Keywords:combinatorial set theory; Proper Forcing Axiom; Ramsey theory; research survey; basis; directed sets; successively binary relations; transitive relations; uncountable linear orderings; uncountable regular spaces; Open Coloring Axiom; Aronszajn trees PDFBibTeX XMLCite \textit{S. Todorcevic}, Doc. Math. Extra Vol., 43--52 (1998; Zbl 0918.03025) Full Text: EuDML EMIS