Problem 70

Let $\mathfrak{c}$ be the ordinal of the real numbers, $\beta$ be any countable ordinal, and $2\leq n<\omega$ . Is it true that $\mathfrak{c}\to (\beta, n)_2^3$ ?

Source: erdosproblems.com/70

Graph theory
Ramsey theory
Set theory

Wiki page

Status

Open

No claim settles this problem.

Open. The site's label is OPEN. One accepted partial claim, Erdős and Rado 1956, settles the instances with $\beta<\omega 2$ and $n\le4$ ; the question stays open.

Er87: Erdős, P., Some problems on finite and infinite graphs. Logic and combinatorics (Arcata, Calif., 1985), Contemp. Math. 65, Amer. Math. Soc. (1987), 223–228; Problem 3, p. 223. Library home: Erdos 1987.
ErRa56: Erdős, P. and Rado, R., A partition calculus in set theory. Bull. Amer. Math. Soc. 62 (1956), no. 5, 427–489; Theorem 31, p. 447. Library home: Erdős and Rado 1956.
Va99: Some of Paul's favorite problems, booklet for the conference "Paul Erdős and his mathematics", Budapest, July 1999; item 7.83, as the site cites it.

Problem 70

Status

Proof claims

Discussion