We present a finite square principle that was used to show the independence of the following statements:
1. If M is a structure in a vocabulary of size ≤λ and D is a regular ultrafilter on λ, then M^λ/D is λ++-universal. (Keisler &
Chang: Open Problem 18)
2. Suppose M and N are structures in a vocabulary of size ≤ λ such that |M|,|N| ≤ λ. If M≡N, D is a regular ultrafilter on
λ, and 2^λ = λ^+, then M^λ/D is isomorphic to N^λ/D.
(Keisler & Chang: Open Problem 19)
Joint work with Saharon Shelah, Jouko Väänänen