On the existence of universal models

Suppose that λ = λ<λ ≥ א0, and we are considering a theory T . We give a criterion on T which is sufficient for the consistent existence of λ++ universal models of T of size λ+ for models of T of size ≤ λ+, and is meaningful when 2λ + > λ++. In fact, we work more generally with abstract elementary classes. The criterion for the consistent existence of universals applies to various well known theories, such as triangle-free graphs and simple theories. Having in mind possible applications in analysis, we further observe that for such λ, for any fixed μ > λ+ regular with μ = μλ + , it is consistent that 2λ = μ and there is no normed vector space over Q of size < μ which is universal for normed vector spaces over Q of dimension λ+ under the notion of embedding h which specifies (a, b) such that ||h(x)||/||x|| ∈ (a, b) for all x.