A Family of Trees with No Uncountable Branches

We construct a family of 2א1 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 2א1 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 2א1 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. 2000 Mathematics Subject Classification. 03E04, 54E99.