On Squares, outside Guessing of Clubs and I <f [λ]

Suppose that λ = μ and μ is singular. We consider two aspects of the square property on subsets of λ. First, we have results which show e.g. that for א0 < κ = cf(κ) < μ, the equality cf([μ],⊆) = μ is a sufficient condition for the set of elements of λ whose cofinality is ≤ κ, to be split into the union of μ sets with squares. Secondly, we introduce a certain weak version of the square property and prove that if μ is a strong limit, then this weak square property holds on λ without any additional assumptions. In the second section we start with two universes V1 ⊆ V2 of set theory, and a regular cardinal κ in V1 such that the cofinality of κ in V2 is θ < κ. Assume κ + is preserved and κ is inaccessible in V1 with 2 κ = κ. We show that then there is an unbounded subset C of κ in V2, such that for every club E of κ in V1, the difference C \ E is bounded. We have further results of a similar flavor. Some of our results were independently obtained by Moti Gitik, using different methods. In the third section we consider the connection between the ideal I[λ] and the notions of square and weak square. We show that these notions are a part of a larger family of properties which all can be introduced through a single definition of I<f [λ] by changing the parameter f . We discuss further properties of I<f [λ] and some other similarly defined notions. We have further results on I[λ] in the last section. 1 Authors partially supported by the Basic Research Foundation Grant number 0327398 administered by the Israel Academy of Sciences and Humanities. The first author thanks the Hebrew University and the Lady Davis Foundation for the Forchheimer Postdoctoral Fellowship. For easier future reference, note that this is publication [DjSh 562] in Shelah’s bibliography. The results presented were obtained in the period April to August 1994. The appendix was added in December 1994. We wish to thank Moti Gitik and Ofer Shafir for their interest and helpful comments, as well as to James Cummings for pointing out a difficulty.