Universality: new criterion for non-existence

Abstract

We find new "reasons" for a class of models for not having a universal model in a cardinal $\lambda$. This work, though it has consequences in model theory, is really in combinatorial set theory. We concentrate on a prototypical class which is a simply defined class of models, of combinatorial character - models of $T_{\rm ceq}$ (essentially another representation of $T_{\rm feq}$ which was already considered but the proof with $T_{\rm ceq}$ is more transparent). Models of $T_{\rm ceq}$ consist essentially of an equivalence relation on one set and a family of choice functions for it. This class is not simple (in the model theoretic sense) but seems to be very low among the non-simple (first order complete countable) ones. We give sufficient conditions for the non-existence of a universal model for it in $\lambda$. This work is continued in [Sh:F2071].