A topological Ramsey classification of countable ordinals. II

We provide optimal values for m satisfying the partition relation ∀l>1,α→(topω2+1)l,m2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\forall l > 1, \alpha \rightarrow ({\rm top} \omega^{2} + 1)^{2}_{l,m}}}$$\end{document} when α=ωω+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\alpha = {\omega^{\omega}} +1}}$$\end{document} and when α=ωωk,foreveryk>1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\alpha = {\omega^{{\omega}^{k}}}, {\rm for every} k > 1}}$$\end{document}.