# Sh:564

- Shelah, S. (1996).
*Finite canonization*. Comment. Math. Univ. Carolin.,**37**(3), 445–456. arXiv: math/9509229 MR: 1426909 -
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}, mrnumber = {1426909}, mrclass = {05D10}, note = {\href{https://arxiv.org/abs/math/9509229}{arXiv: math/9509229}}, arxiv_number = {math/9509229} }