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Ramsey theory; Erdös-Rado theorem; canonization
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 $[{1,\dotsc,m^*}]^n$, for some $A \in [{1,\dotsc,m^*}]^m$, the question of when the equality $f({i_1,\dotsc,i_n}) = f({j_1,\dotsc,j_n})$ (where $i_1 < \cdots < i_n$ and $j_1 < \cdots j_n$ are from $A$) holds has the simplest answer: for some $v \subseteq \{1,\dotsc,n\}$ the equality holds iff $\bigwedge_{\ell \in v} i_\ell = j_\ell$. 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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