Erdős Problem 70: Let $\mathfrak{c}$ be the cardinality of the continuum, let $\beta$ be a countable ordinal, and let $2 \le n < \omega$. Is it true that $\mathfrak{c} \to (\beta, n)^3_2$?

First open case beyond Erdős–Rado: $\mathfrak{c} \to (\omega \cdot 2, 4)^3_2$.

Erdős and Rado proved $\mathfrak{c} \to (\omega + n, 4)^3_2$ for every finite $n \ge 2$ (see erdos_rado), which covers all red ordinals below $\omega \cdot 2 = \omega + \omega$. This variant asks whether the result extends to $\beta = \omega \cdot 2$, the simplest countable ordinal not covered by their theorem.