$\omega _ 3\omega _ 1\to (\omega _ 3\omega _ 1,3)^ 2$ requires an inaccessible

Abstract:

We show that if there is a simplified $({\omega _2},1)$-morass with linear limits and ${2^{{\aleph _1}}} = {\aleph _2}$, then ${\omega _3}{\omega _1} \nrightarrow {({\omega _3}{\omega _1},3)^2}$. Thus, assuming ${2^{{\aleph _1}}} = {\aleph _2}$, this negative relation holds in $V$ if both ${\aleph _2}$ and ${\aleph _3}$ are (successor cardinals)$^{L}$, since in this case, well-known arguments show there is a simplified $({\omega _2},1)$-morass with linear limits. The contrapositive is that, assuming ${2^{{\aleph _1}}} = {\aleph _2}$, the positive relation holds only if either ${\aleph _2}$ or ${\aleph _3}$ is (inaccessible)$^{L}$.