A note on the Canonical Ramsey Theorem and Ramsey ultrafilters
We give a characterizations of Ramsey ultrafilters on $\mathscr P(ω)$ in terms of functions $f:ω^n\toω$ and their ultrafilter extensions. To do this, we prove that for any partition $\mathcal P$ of $[ω]^n$ there is a finite partition~$\mathcal Q$ of $[ω]^{2n}$ such that any set $X\subseteq ω$ that is homogeneous for $\mathcal Q$ is a finite union of sets that are canonical for $\mathcal P$.
math.LO↗