Finite canonization

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, ...,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 l in v)(i_l = j_l). 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.