Abstract

We strengthen nonstructure theorems for almost free Abelian groups by studying long Ehrenfeucht–Fraı̈ssé games between a fixed group of cardinality λ and a free Abelian group. A group is called ε -game-free if the isomorphism player has a winning strategy in the game of length ε ∈ λ . We prove for a large set of successor cardinals λ = μ + the existence of nonfree -game-free groups of cardinality λ . We concentrate on successors of singular cardinals