### Finite Canonization

by Shelah. [Sh:564]

Commentationes Math Universitatis Carolinae, 1996

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, ...,m^*}]^n, for some
A in [{1, ...,m^*}]^m, the question of when the equality
f({i_1, ...,i_n})=f({j_1, ...,j_n}) (where i_1< ... <i_n
and j_1 < ... < j_n are from A) holds has the simplest answer:
for some v subseteq {1, ...,n} the equality holds iff
(for all 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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