# Sh:614

- Džamonja, M., & Shelah, S. (2004).
*On the existence of universal models*. Arch. Math. Logic,**43**(7), 901–936. arXiv: math/9805149 DOI: 10.1007/s00153-004-0235-1 MR: 2096141 -
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 aplpications 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 {\mathbb Q} of size <\mu which is universal for normed vector spaces over {\mathbb Q} of dimension \lambda^+ under the notion of embedding h which specifies (a,b) such that \|h(x)\|/\|x\|\in (a,b) for all x.

- published version (36p)

Bib entry

@article{Sh:614, author = {D{\v{z}}amonja, Mirna and Shelah, Saharon}, title = {{On the existence of universal models}}, journal = {Arch. Math. Logic}, fjournal = {Archive for Mathematical Logic}, volume = {43}, number = {7}, year = {2004}, pages = {901--936}, issn = {0933-5846}, doi = {10.1007/s00153-004-0235-1}, mrclass = {03E35 (03C50 03C55 03C95)}, mrnumber = {2096141}, mrreviewer = {O. V. Belegradek}, doi = {10.1007/s00153-004-0235-1}, note = {\href{https://arxiv.org/abs/math/9805149}{arXiv: math/9805149}}, arxiv_number = {math/9805149} }