An AEC satisfying the disjoint amalgamation property, has arbitrarily   large models
  We study AECs without assuming the amalgamation property in general. We do assume the disjoint amalgamation property in a specific cardinality lambda and assume that there is no maximal model in \lambda.   Under these hypotheses, we prove the following: 1. for every model, M, of cardinality \lambda, and every \mu>\lambda, we can find a model M^* of cardinality \mu, extending M. 2.(\lambda,\lambda,\mu)-amalgalmation property: for every three models M,N,M^* of cardinalities \lambda,\lambda,\mu, respectively, if M<M^* and M<N then we can amalgamate M^* and N over M. 
Adi Jarden
math.LO