### Almost free groups and Ehrenfeucht-Fra{\"\i}ss\'e games for successors of singular cardinals

by Shelah and Vaisanen. [ShVs:787]

Annals Pure and Applied Logic, 2002

We strengthen non-structure theorems for almost free Abelian
groups by studying long Ehrenfeucht-Fra{i}sse games between a
fixed group of cardinality lambda and a free Abelian group.
A group is called epsilon-game-free if the isomorphism player has
a winning strategy in the game (of the described form) of length
epsilon in lambda . We prove for a large set of successor
cardinals lambda = mu^+ existence of nonfree (mu .
omega_1)-game-free groups of cardinality lambda . We concentrate
on successors of singular cardinals.

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