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 . We use versions of being reduced—replacing by a subring (defined by a sequence )—and get quite accurate results for the existence of universals in a cardinal, for embeddings and for pure embeddings. Second, we deal with (variants of) the oak property (from a work of Džamonja and the author), 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 (naturally interpreted, so for quantifier-free formulas) and deals more with the existence of universals.
"Universal Structures." Notre Dame J. Formal Logic 58 (2) 159 - 177, 2017. https://doi.org/10.1215/00294527-3800985