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

Quotients of Strongly Proper Forcings and Guessing Models

We prove that a wide class of strongly proper forcing posets have quotients with strong properties. Specifically, we prove that quotients of forcing posets which have simple universal strongly generic conditions on a stationary set of models by certain nice regular suborders satisfy the $\omega_1$-approximation property. We prove that the existence of stationarily many $\omega_1$-guessing models in $P_{\omega_2}(H(\theta))$, for sufficiently large cardinals $\theta$, is consistent with the continuum being arbitrarily large, solving a problem of Viale and Weiss

On properties of theories which preclude the existence of universal models

We introduce the oak property of first order theories, which is a syntactical condition that we show to be sufficient for a theory not to have universal models in cardinality ? when certain cardinal arithmetic assumptions about ? implying the failure of GCH (and close to the failure of SCH) hold. We give two examples of theories that have the oak property and show that none of these examples satisfy SOP4, not even SOP3. This is related to the question of the connection of the property SOP4 to non-universality, as was raised by the earlier work of Shelah. One of our examples is the theory View the MathML source for which non-universality results similar to the ones we obtain are already known; hence we may view our results as an abstraction of the known results from a concrete theory to a class of theories. We show that no theory with the oak property is simple

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