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Claim. Theorem 31 of Erdős and Rado, A partition calculus in set theory, states in its relation (30) (p. 447): if $\phi$ is an order type with $\lvert\phi\rvert>\aleph_0$ into which neither $\omega_1$ nor its converse $\omega_1^*$ embeds, then $\phi\to(4,\alpha)^3$ for every $\alpha<\omega_0 2$. The order type $\lambda$ of the real line meets the hypothesis: it is uncountable, and every well-ordered or conversely well-ordered set of reals is countable. With the two colors exchanged, $\lambda\to(\beta,4)^3_2$ for every $\beta<\omega 2$: every two-coloring of the triples of reals has either a set of reals of order type $\beta$ all of whose triples have the first color or a set of four reals all of whose triples have the second. The relation $\mathfrak c\to(\omega+n,4)^3_2$ that the site's commentary credits to Erdős and Rado is the case $\beta=\omega+n$, and Erdős [Er87] calls it an old result of Rado and himself. The source card is erdos_1956_partition_calculus_set_theory.

Covers. The instances $(\beta,n)$ of Problem 70 with $\beta<\omega 2$ and $n\le4$. The theorem says nothing about the instances with $\beta\ge\omega 2$ and $n\ge4$, of which $(\omega 2,4)$ is the first, nor about $n\ge5$ with $\omega<\beta<\omega 2$.

Depends on. No page of this wiki; the result rests on the refereed paper linked above.

Acceptance. Refereed: Bull. Amer. Math. Soc. 62 (1956), no. 5, 427–489, received by the editors on 17 May 1955; Theorem 31 is on p. 447 and the proof of (30) on pp. 454–457. The page is dated by the September 1956 issue, which gives no day, so the first of the month stands in for it. The site credits the result in its commentary but labels the problem OPEN, so the commentary is not reviewed evidence. No formal proof of the relation is recorded, so no formalized evidence is listed. Nothing here is this project's own review.