On the existence of universal models

Suppose that λ=λ <^λ א ₀, 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 2^{λ +}>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=μ ^λ+, it is consistent that 2^λ=μ and there is no normed vector space over Q of size <μ which is universal for normed vector spaces over < of dimension λ⁺ under the notion of embedding h which specifies (a,b) such that ∥h(x)∥/∥x∥∈ (a,b) for all x.