More on regular reduced products

The authors show, by means of a finitary version ¤ λ,D of the combinatorial principle ¤ ∗ λ of [7], the consistency of the failure, relative to the consistency of supercompact cardinals, of the following: for all regular filters D on a cardinal λ, if Mi and Ni are elementarily equivalent models of a language of size ≤ λ, then the second player has a winning strategy in the Ehrenfeucht-Fräıssé game of length λ+ on ∏ i Mi/D and ∏ i Ni/D. If in addition 2 λ = λ+ and i < λ implies |Mi|+ |Ni| ≤ λ + this means that the ultrapowers are isomorphic. This settles negatively conjecture 18 in [2]. The problem of when two elementarily equivalent structures have isomorphic ultrapowers was prominent in the model theory of the 1960’s. Keisler [3] proved, assuming GCH, that elementarily equivalent structures have isomorphic ultrapowers. Keisler’s proof depended on GCH both on the question of existence of good ultrafilters and on limiting the size of the ultraproducts. More exactly, Keisler considered a language of size λ, models M of size ≤ λ and a λ-good countably incomplete ultrafilter D on λ. He proved that M/D is λ-saturated. Under the weaker assumption that D is regular he Research partially supported by grant 40734 of the Academy of Finland. The second author would like to thank the Israel Science Foundation for partial support of this research (Grant no. 242/03). Publication 852.