On the existence of universal models

Abstract

Suppose that $\lambda=\lambda^{<\lambda} \ge\aleph_0$, 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 ${\Bbf Q}$ of size $<\mu$ which is universal for normed vector spaces over ${\Bbf Q}$ of dimension $\lambda^+$ under the notion of embedding $h$ which specifies $(a,b)$ such that $\norm{h(x)}/\norm{x}\in (a,b)$ for all $x$.