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on Erdős #1167 · set theory / ramsey theory

The statement

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.

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