On a problem of Erdos and Hajnal

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 $\aleph_{\omega}$.