On a problem of Erdos and Hajnal

Abstract

We address a question of Erdős and Hajnal about the ordinary partition relation $\aleph_{ω+1}\nrightarrow(\aleph_{ω+1},(3)_{\aleph_0})^2$. For $θ=\mathrm{cf}(λ)<λ$, assuming $2^λ=λ^+$ they proved the negative relation $λ^+\nrightarrow(λ^+,(3)_θ)^2$ and asked whether the (local instance of) GCH is indispensable. We show that this negative relation is consistent with $λ$ being a strong limit and $2^λ>λ^+$. The result can be pushed down to $\aleph_ω$.