A characterization of the finite model property for $\omega$-categorical theories

Abstract

A first-order theory $T$ has the {\em finite model property} if every first-order sentence that is implied by $T$ has a finite model. We give a characterization of when a complete $\omega$-categorical theory $T$ has the finite model property. The characterization makes it possible to transfer stability theoretic results in [29] to certain classes of finite structures. This approach also gives information concerning properties that a sequence of finite ``base structures'' of ever larger size must have in order for its random expansions to a larger vocabulary to have a chance to satisfy a convergence law for first-order logic.