OPEN

Let $\mathfrak{c}$ be the ordinal of the real numbers, $\beta$ be any countable ordinal, and $2\leq n<\omega$. Is it true that $\mathfrak{c}\to (\beta, n)_2^3$?

Erdős and Rado proved that $\mathfrak{c}\to (\omega+n,4)_2^3$ for any $2\leq n<\omega$.