On a problem of Erdos and Hajnal

Abstract

We address a question of Erdős and Hajnal about the ordinary partition relation ${\aleph }_{\omega +1}\nrightarrow ({\aleph }_{\omega +1},(3{)}_{{\aleph }_0}{)}^2$. For $\theta =\mathrm{cf}(\lambda )<\lambda$, assuming $2^{\lambda }={\lambda }^{+}$ they proved the negative relation ${\lambda }^{+}\nrightarrow ({\lambda }^{+},(3{)}_{\theta }{)}^2$ and asked whether the (local instance of) GCH is indispensable. We show that this negative relation is consistent with $\lambda$ being a strong limit and $2^{\lambda }>{\lambda }^{+}$. The result can be pushed down to ${\aleph }_{\omega }$.