On the existence of universal models

Abstract

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