A club guessing toolbox I

Abstract

Club guessing principles were introduced by Shelah as a weakening of Jensen's diamond. Most spectacularly, they were used to prove Shelah's ZFC bound on the power of the first singular cardinal. These principles have found many other applications: in cardinal arithmetic and PCF theory; in the construction of combinatorial objects on uncountable cardinals such as Jonsson algebras, strong colourings, Souslin trees, and pathological graphs; to the non-existence of universals in model theory; to the non-existence of forcing axioms at higher uncountable cardinals; and many more. In this paper, the first part of a series, we survey various forms of club-guessing that have appeared in the literature, and then systematically study the various ways in which a club-guessing sequences can be improved, especially in the way the frequency of guessing is calibrated. We include an expository section intended for those unfamiliar with club-guessing and which can be read independently of the rest of the article.

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