On the existence of universal models

Suppose that $\lambda ={\lambda }^{<\lambda }\ge {\mathrm{\aleph }}_0$, and we are considering a theory $T$. We give a criterion on $T$ which is sufficient for the consistent existence of ${\lambda }^{++}$ universal models of $T$ of size ${\lambda }^{+}$ for models of $T$ of size $\le {\lambda }^{+}$, and is meaningful when $2^{{\lambda }^{+}}>{\lambda }^{++}$. 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 $\lambda$, for any fixed $\mu >{\lambda }^{+}$ regular with $\mu ={\mu }^{{\lambda }^{+}}$, it is consistent that $2^{\lambda }=\mu$ and there is no normed vector space over $\text{Bbf}Q$ of size $<\mu$ which is universal for normed vector spaces over $\text{Bbf}Q$ of dimension ${\lambda }^{+}$ under the notion of embedding $h$ which specifies $(a,b)$ such that $\norm{h(x)}/\norm{x}\in (a,b)$forall$x$.