We discuss the relationships between the notions of $\kappa $ -cub game on $\lambda $ , $\kappa $ -cub subset of $\lambda $ , the ideal of good subsets of $\lambda $ and the problem of adding a $\kappa $ -cub into a given $\kappa $ -stationary subset of $\lambda $ . We also give a short introduction to the ideal of good subsets of $\lambda $.

We show in the paper that for any non-classifiable countable theory T there are non-isomorphic models and that can be forced to be isomorphic without adding subsets of small cardinality. By making suitable cardinal arithmetic assumptions we can often preserve stationary sets as well. We also study non-structure theorems relative to the Ehrenfeucht-Fraïssé game.

We generalize the result of Mekler and Shelah [3] that the existence of a canary tree is independent of ZFC + GCH to uncountable regular cardinals. We also correct an error from the original proof.

We characterize the classifiability of a countable first-order theory T in terms of the solvability of the potential-isomorphism problem for models of T.

We generalize the result of Mekler and Shelah [3] that the existence of a canary tree is independent of ZFC + GCH to uncountable regular cardinals. We also correct an error from the original proof.