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Published online by Cambridge University Press: 12 March 2014
The authors show, by means of a finitary version of the combinatorial principle
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 ΠiMi/D and ΠiNi/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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