Let K be an abstract elementary class and \lambda=LS(K). K can be axiomatized by a sentence in L_{(2^\lambda)^+,\lambda^+}, allowing game quantification. This is parallel to Shelah-Villaveces result which demands a much higher complexity of junctions but without game quantification. Shelah’s presentation theorem gives K = PC(T, \Gamma, L(K)) where T is a first-order theory in an expansion of L(K) and \Gamma is a set of T-types. We provide a better bound of |\Gamma| in terms of I_2(\lambda,K). We also give conditions under which the categoricity in two successive cardinals implies the existence of models in the next cardinal. This improves the result of Shelah.