A note on the two cardinal problem

In this note we prove a theorem concerning the two cardinal problem (see [1 ], [4], [6], [7] for reference and for some of the standard notation); this result has been referred to in [6, p. 311] and [7, (3.7) ]. The problem, first proposed by Vaught, is as follows. Let T be a first-order theory and let U be a unary predicate symbol in the language of T. T is said to admit the pair a, ( of cardinals if there exists a model M=(A, U, S, * * * ) of T such that |A| =a and I U|I =j3. Suppose T admits a pair a, ( where a >l >?w. Then what other pairs of cardinals y, 5 must T admit? The following theorem gives a partial answer.