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on The 3-uniform partition relation at the first uncountable ordinal

The statement

Let $\mathfrak{c}$ denote the cardinality of the continuum, and let $\omega_1 = \aleph_1$ denote the first uncountable ordinal. For each integer $n \geq 2$ , consider the 3-uniform ordinal Ramsey property $\mathfrak{c} \to (\omega_1, n)^3_2$ , defined as follows. For any 2-coloring of all 3-element subsets of the ordinal type of $\mathfrak{c}$ (where the coloring is given as a predicate on ordered triples of distinct elements that is invariant under permutation of its arguments), either there exists a red-monochromatic subset of order type $\omega_1$ (every 3-element subset of which is colored red), or there exists a blue-monochromatic subset of cardinality $n$ (every 3-element subset of which is colored blue). Determine whether $\mathfrak{c} \to (\omega_1, n)^3_2$ holds for all integers $n \geq 2$ .

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.