  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].
