# Sh:583

- Gilchrist, M., & Shelah, S. (1997).
*The consistency of ZFC + 2^{\aleph_0}>\aleph_\omega+\mathcal I(\aleph_2)=\mathcal I(\aleph_\omega)*. J. Symbolic Logic,**62**(4), 1151–1160. arXiv: math/9603219 DOI: 10.2307/2275632 MR: 1617993 -
Abstract:

An \omega-coloring is a pair \langle f,B\rangle where f:[B]^{2}\longrightarrow\omega. The set B is the field of f and denoted Fld(f). Let f,g be \omega-colorings. We say that f realizes the coloring g if there is a one-one function k:Fld(g)\longrightarrow Fld(f) such that for all \{x,y\}, \{u,v\}\in dom(g) we have f(\{k(x),k(y)\})\neq f(\{k(u),k(v)\}) \Rightarrow g(\{x,y\})\neq g(\{u,v\}). We write f\sim g if f realizes g and g realizes f. We call the \sim-classes of \omega-colorings with finite fields identities. We say that an identity I is of size r if |Fld(f)|=r for some/all f\in I. For a cardinal \kappa and f:[\kappa]^2\longrightarrow\omega we define {\mathcal I}(f) to be the collection of identities realized by f and {\mathcal I }(\kappa) to be \bigcap\{{\mathcal I}(f)| f:[\kappa]^2\longrightarrow\omega\}.We show that, if ZFC is consistent then ZFC + 2^{\aleph_0}>\aleph_\omega + {\mathcal I}(\aleph_2)={\mathcal I}(\aleph_\omega) is consistent.

- Current version: 1996-03-16_10 (13p) published version (10p)

Bib entry

@article{Sh:583, author = {Gilchrist, Martin and Shelah, Saharon}, title = {{The consistency of ZFC + $2^{\aleph_0}>\aleph_\omega+\mathcal I(\aleph_2)=\mathcal I(\aleph_\omega)$}}, journal = {J. Symbolic Logic}, fjournal = {The Journal of Symbolic Logic}, volume = {62}, number = {4}, year = {1997}, pages = {1151--1160}, issn = {0022-4812}, mrnumber = {1617993}, mrclass = {03E35}, doi = {10.2307/2275632}, note = {\href{https://arxiv.org/abs/math/9603219}{arXiv: math/9603219}}, arxiv_number = {math/9603219} }