OPEN
This is open, and cannot be resolved with a finite computation.
Let $r\geq 2$ be finite and $\lambda$ be an infinite cardinal. Let $\kappa_\alpha$ be cardinals for all $\alpha<\gamma$.
Is it true that\[2^\lambda \to (\kappa_\alpha+1)_{\alpha<\gamma}^{r+1}\]implies\[\lambda \to (\kappa_\alpha)_{\alpha<\gamma}^{r}?\]Here $+$ means cardinal addition, so that $\kappa_\alpha+1=\kappa_\alpha$ if $\kappa_\alpha$ is infinite.
A problem of Erdős, Hajnal, and Rado.
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This page was last edited 23 January 2026.
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