A two-cardinal theorem

Abstract:

We prove the following theorem and deal with some related questions: If for all $n < \omega ,T$ has a model $M$ such that ${n^n} \leq |{Q^M}{|^n} \leq |{P^M}| < {\aleph _0}$ then for all $\lambda ,\mu$ such that $|T| \leq \mu \leq \lambda < {\operatorname {Ded} ^ \ast }(\mu )$ (e.g. $\mu = {\aleph _0},\lambda = {2^{{\aleph _0}}}), T$ has a model of type $(\lambda ,\mu )$, i.e. $|{Q^M}| = \mu ,|{P^M}| = \lambda$.