Disjoint Amalgamation in Locally Finite AEC

We introduce the concept of a locally finite abstract elementary class and develop the theory of excellence (with respect to disjoint ( , k)-amalgamation) for such classes. From this we find a family of complete L!1,! sentences r that a) homogeneously characterizes @r (improving results of Hjorth [12] and Laskowski-Shelah [14] and answering a question of [22]), while b) the r provide the first examples of a class of models of a complete sentence in L!1,! where the spectrum of cardinals in which amalgamation holds is other that none or all.