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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 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 $\aleph_ω$.

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Shimon Garti, Yair Hayut, Saharon Shelah. 2026-06-25. On a problem of Erdos and Hajnal. https://arxiv.org/abs/2502.16625

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