We show that many singular cardinals λ above a strongly compact cardinal have regular ultrafilters D that violate the finite square principle □finλ,D introduced in . For such ultrafilters D and cardinals λ there are models of size λ for which Mλ/D is not λ++-universal and elementarily equivalent models M and N of size λ for which Mλ/D and Nλ/D are non-isomorphic. The question of the existence of such ultrafilters and models was raised in .
"Regular ultrafilters and finite square principles." J. Symbolic Logic 73 (3) 817 - 823, September 2008. https://doi.org/10.2178/jsl/1230396748