For many classes of models, there are universal members in any cardinal λ which “essentially satisfies GCH, i.e., λ = 2<λ,” in particular for the class of models of a complete first-order T (well, if at least λ > |T|). But if the class is “complicated enough”, e.g., the class of linear orders, we know that if λ is “regular and not so close to satisfying GCH”, then there is no universal member. Here, we find new sufficient conditions (which we call the olive property), not covered by earlier cases (i.e., fail the so-called SOP4). The advantage of those conditions is witnessed by proving that the class of groups satisfies one of those conditions.
Funding source: Israel Science Foundation
Award Identifier / Grant number: 1053/11
Funding statement: The author thanks the Israel Science Foundation for partial support of this work (Grant No. 1053/11, Publication 1029).
The author thanks Alice Leonhardt for the beautiful typing. We would also like to thank the referee and Thilo Weinert for helpful comments.
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