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arXiv · 2502.16625

On a problem of Erdos and Hajnal

Shimon Garti·Yair Hayut·Saharon Shelah
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Abstract

We address a question of Erd\H{o}s 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 $\aleph_{\omega}$.

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Shimon Garti, Yair Hayut, Saharon Shelah. 2025-02-23. On a problem of Erdos and Hajnal. https://arxiv.org/abs/2502.16625

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