Almost free groups and Ehrenfeucht–Fraı̈ssé games for successors of singular cardinals Original Research Article

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 the described form) of length ε∈λ. We prove for a large set of successor cardinals λ=μ+ the existence of nonfree (μ·ω1)-game-free groups of cardinality λ. We concentrate on successors of singular cardinals.