# 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.
• 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.

• 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}
}