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 $\mathfrak{c} \to (\beta, n)^3_2$?

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 $\mathfrak{c} \to (\beta, n)^3_2$?

Note: The cases $n \le 3$ are trivially true (see omega_three), so the genuine content of the conjecture begins at $n = 4$.