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 λ when certain cardinal arithmetic assumptions about λ implying the failure of GCH 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 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. (shrink)

We discuss dividing lines (in model theory) and some test questions, mainly the universality spectrum. So there is much on conjectures, problems and old results, mainly of the author and also on some recent results.

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.

We deal with the existence of universal members in a given cardinality for several classes. First, we deal with classes of abelian groups, specifically with the existence of universal members in cardinalities which are strong limit singular of countable cofinality or λ=λℵ0. We use versions of being reduced—replacing Q by a subring —and get quite accurate results for the existence of universals in a cardinal, for embeddings and for pure embeddings. Second, we deal with the oak property, a property of (...) complete first-order theories sufficient for the nonexistence of universal models under suitable cardinal assumptions. Third, we prove that the oak property holds for the class of groups and deals more with the existence of universals. (shrink)