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 $\aleph_{\omega}$.
From: Yair Hayut [view email]
[v1]
Sun, 23 Feb 2025 15:59:58 UTC (22 KB)
[v2]
Thu, 25 Jun 2026 18:15:25 UTC (33 KB)
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