The authors show, by means of a finitary version □finλ, D of the combinatorial principle □b*λ of , 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-Fraïssé game of length λ+ on ∏i Mi/D and ∏i Ni/D. If in addition 2λ=λ+ and i <λ implies |Mi|+|Ni|≤ λ+ this means that the ultrapowers are isomorphic. This settles negatively conjecture 18 in .
"More on regular reduced products." J. Symbolic Logic 69 (4) 1261 - 1266, December 2004. https://doi.org/10.2178/jsl/1102022222