Club guessing and the universal models

We survey the use of club guessing and other pcf constructs in the context of showing that a given partially ordered class of objects does not have a largest, or a universal element

A family of trees with no uncountable branches

We construct a family of 2 ℵ1 trees of size ℵ1 and no uncountable branches that in a certain way codes all ω1sequences of infinite subsets of ω. This coding allows us to conclude that in the presence of the club guessing between ℵ1 and ℵ0, these trees are pairwise very different. In such circumstances we can also conclude that the universality number of the ordered class of trees of size ℵ1 with no uncountable branches under “metric-preserving ” reductions must be at least the continuum. From the topological point of view, the above results show that under the same assumptions there are 2 ℵ1 pairwise non-isometrically embeddable first countable ω1metric spaces with a strong non-ccc property, and that their universality number under isometric embeddings is at least the continuum. Without the non-ccc requirement, a family of 2 ℵ1 pairwise non-isometrically embeddable first countable ω1-metric spaces exists in ZFC by an earlier result of S. Todorčević. The set-theoretic assumptions mentioned above are satisfied in many natural models of set theory (such as the ones obtained after forcing by a ccc forcing over a model of ♦). We use a similar method to discuss trees of size κ with no uncountable branches, for any regular uncountable κ

Saturated filters at successors of singulars, weak reflection and yet another weak club principle

Suppose that lambda is the successor of a singular cardinal mu whose cofinality is an uncountable cardinal kappa. We give a sufficient condition that the club filter of lambda concentrating on the points of cofinality kappa is not lambda^+-saturated. The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflectio