The weak non-finite cover property (wnfcp) was introduced in [1] in connection with “axiomatizability” of
lovely pairs of models of a simple theory.
We find a combinatorial condition on a simple theory equivalent to the wnfcp, yielding a direct proof that the non-finite
cover property implies the wnfcp, and that the wnfcp is preserved under reducts. We also study the question whether the
wnfcp is preserved when passing from
a simple theory T to the theory TP of lovely pairs of models of T (true in the stable case). While the question
remains open, we show, among other things, that if (for a T with the wnfcp) TP is low, then TP has the wnfcp. To
study this question, we describe “double lovely pairs”, and, along the way, we develop the notion of a “lovely
n-tuple” of models of a simple theory, which is an analogue of the notion of a beautiful tuple of models
of stable theories [2].