More on regular reduced products

Abstract

The authors show, by means of a finitary version □^fin_{λ, D} of the combinatorial principle □^b*_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 M_i and N_i are elementarily equivalent models of a language of size ≤ λ, then the second player has a winning strategy in the Ehrenfeucht-Fraïssé game of length λ⁺ on ∏_i M_i/D and ∏_i N_i/D. If in addition 2^λ=λ⁺ and i <λ implies |M_i|+|N_i|≤ λ⁺ this means that the ultrapowers are isomorphic. This settles negatively conjecture 18 in [2].