A TEST FOR EXPANDABILITY

A model M of countable similarity type and cardinality kappa is expand able if every consistent extension T-1 of its complete theory with /T- 1/ less than or equal to kappa is satisfiable in M and it is compactly expandable if every such extension which additionally is finitely sat isfiable in M is satisfiable in M. In the countable case and in the ca se of a model of cardinality greater than or equal to 2(omega) of a su perstable theory without the finite cover property the notions of satu ration, expandability and compactness for expandability agree. The que stion of the existence of compactly expandable models which are not ex pandable is open. Here we present a test which serves to prove that a compactly expandable model of cardinality greater than or equal to 2(o mega) of a superstable theory is expandable. It is stated in terms of the existence of a certain elementary submodel whose corresponding the ory of pairs of models satisfies a weak elimination of Ramsey quantifi ers.