# Sh:564

• Shelah, S. (1996). Finite canonization. Comment. Math. Univ. Carolin., 37(3), 445–456.
• Abstract:
The canonization theorem says that for given m,n for some m^* (the first one is called ER(n;m)) we have: for every function f with domain [{1,\ldots,m^*}]^n, for some A\in [{1,\ldots,m^*}]^m, the question of when the equality f({i_1,\ldots,i_n})=f({j_1,\ldots,j_n}) (where i_1<\ldots<i_n and j_1 <\ldots< j_n are from A) holds has the simplest answer: for some v \subseteq \{1,\ldots,n\} the equality holds iff (\forall\ell\in v)(i_\ell = j_\ell).

In this paper we improve the bound on ER(n,m) so that fixing n the number of exponentiation needed to calculate ER(n,m) is best possible.

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Bib entry
@article{Sh:564,
author = {Shelah, Saharon},
title = {{Finite canonization}},
journal = {Comment. Math. Univ. Carolin.},
fjournal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {37},
number = {3},
year = {1996},
pages = {445--456},
issn = {0010-2628},
mrclass = {05D10},
mrnumber = {1426909},
mrreviewer = {Jaroslav Ne\v{s}et\v{r}il},
note = {\href{https://arxiv.org/abs/math/9509229}{arXiv: math/9509229}},
arxiv_number = {math/9509229}
}