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 λ. Under these hypotheses, we prove the following: 1. for every model, M, of cardinality λ, and every μ>λ, we can find a model M^* of cardinality μ, extending M. 2.(λ,λ,μ)-amalgalmation property: for every three models M,N,M^* of cardinalities λ,λ,μ, respectively, if M<M^* and M<N then we can amalgamate M^* and N over M.

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