  In earlier work of the second and third author the equivalence of a finite
square principle square^fin_{lambda,D} with various model theoretic properties
of structures of size lambda and regular ultrafilters was established. In this
paper we investigate the principle square^fin_{lambda,D}, and thereby the above
model theoretic properties, at a regular cardinal. By Chang's Two-Cardinal
Theorem, square^fin_{lambda,D} holds at regular cardinals for all regular
filters D if we assume GCH. In this paper we prove in ZFC that for certain
regular filters that we call "doubly^+ regular", square^fin_{lambda,D} holds at
regular cardinals, with no assumption about GCH. Thus we get new positive
answers in ZFC to Open Problems 18 and 19 in the book "Model Theory" by Chang
and Keisler.
