Abstract

Suppose that $\alpha = \gamma + \delta$ where $\gamma \geq \delta > 0$. Then there is a graph $\mathscr{G} = $ which has no independent set of order type $\omega^{\omega^\alpha}$ and has no pentagram. In the notation of Erdos and Rado, who generalized Ramsey's Theorem to this setting, $\omega^{\omega^\alpha} \nrightarrow ^2.$