Erdős Problem #1167

**Erdős Problem 1167.** Let \(r \geq 2\) be finite, \(\gamma \geq 2\), and \(\lambda\) be an infinite cardinal. Let \(\kappa_\alpha > r\) 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.