Erdős Problem 1167. Let $r \geq 2$ be finite, $\gamma \geq 2$, 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.

Erdős Problem 1167. Let $r \geq 2$ be finite, $\gamma \geq 2$, 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.