The partition relation $\mathfrak{c}\to(\beta,n)^3_2$ for countable $\beta$ and finite $n$ (Erdős #70)

Write $\mathfrak{c}$ for the cardinality of the continuum. The partition arrow $\kappa\to(\lambda,\mu)^r_2$ means: for every colouring of the $r$-element subsets of a set of order type $\kappa$ with $2$ colours (say red and blue), there is either a subset of order type $\lambda$ all of whose $r$-subsets are red, or a subset of order type $\mu$ all of whose $r$-subsets are blue. Let $\beta$ be any countable ordinal and let $2\leq n<\omega$ be finite. Is it true that $$\mathfrak{c}\to(\beta,n)^3_2\,?$$ Equivalently: must every $2$-colouring of the triples ($3$-element subsets) of a set of size continuum contain either a red-homogeneous subset of order type $\beta$ or a blue-homogeneous subset of size $n$?

Acceptance. FULLY RESOLVES: a complete proof, for all countable ordinals $\beta$ and all finite $n\geq 2$, that $\mathfrak{c}\to(\beta,n)^3_2$ holds — or a proof that it fails for some countable $\beta$ and finite $n$ (an explicit or forced $2$-colouring of the triples of a set of size continuum with no red subset of order type $\beta$ and no blue subset of size $n$) — with all set-theoretic hypotheses (ZFC, CH, or an independence result) clearly flagged. A machine-checkable proof (Lean/Coq) is preferred, since a formalised statement already exists; otherwise a full rigorous written proof. ADVANCES: extend the Erdős–Rado result $\mathfrak{c}\to(\omega+n,4)^3_2$ to a strictly larger countable ordinal on the infinite side (e.g. establish $\mathfrak{c}\to(\omega\cdot 2,n)^3_2$, or $\mathfrak{c}\to(\beta,n)^3_2$ for a new class of countable $\beta$) or to the full range of finite $n$, with a complete proof; or a consistency/independence result pinning down when the relation can fail. Deliver the proof file or written proof, with all set-theoretic hypotheses stated.