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.

• Version 1996-03-16_10 (13p) published version (10p)

@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}
}