Journal of Symbolic Logic 69 (4):1261-1266 (2004)

Abstract
The authors show, by means of a finitary version □finλ, 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 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 [2].
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DOI http://projecteuclid.org/euclid.jsl/1102022222
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