The 3-uniform partition relation for the continuum

Erdős Problem 70: Let $\mathfrak{c}$ denote the cardinality of the continuum. For ordinals $\alpha$, $\beta$ and a cardinal $c$, the 3-uniform ordinal Ramsey property $\alpha \to (\beta, c)^3_2$ is defined as follows: for every 2-coloring of the 3-element subsets of $\alpha$ (identified with its ordinal type), either there exists a subset of order type $\beta$ all of whose 3-element subsets are colored red, or there exists a subset of cardinality $c$ all of whose 3-element subsets are colored blue. The coloring is formally given as a symmetric predicate on ordered triples of distinct elements. Determine whether for every countable ordinal $\beta$ and every integer $n \geq 2$, the partition relation $\mathfrak{c} \to (\beta, n)^3_2$ holds.