Problem 70

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claims/: The 1 claim page of Problem 70, one per claimant's result; the problem's standing derives from them.

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

Formulation. The statement's "ordinal of the real numbers" is read as the order type $\lambda$ of the real line with its usual order, not as the initial ordinal of the cardinal $\mathfrak c$, because that is how the source reads it. Erdős [Er87, Problem 3, p. 223] poses the question as an extension of $\mathfrak c\to(\omega+n,4)^3_2$, which he calls an old result of Rado and himself; that result is Theorem 31 of [ErRa56], proved for the uncountable order types into which neither $\omega_1$ nor $\omega_1^*$ embeds, a hypothesis the real line meets and the initial ordinal of the continuum does not. The site's commentary credits the same relation. The statement in formal-conjectures poses the main question on $\mathbb R$ with its usual order, after a correction of 2026-09-12, while its variant omega_three uses the initial ordinal of the continuum.

Status. 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.

Source. erdosproblems.com/70, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #70, https://www.erdosproblems.com/70.

References.

• [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_problems_finite_infinite_graphs.
• [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: erdos_1956_partition_calculus_set_theory.
• [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.

Formalization. Statement in formal-conjectures.

Current assessment

The question, in the site's formulation read as the Formulation says, asks whether $\lambda\to(\beta,n)^3_2$ for every countable ordinal $\beta$ and every finite $n\ge2$, where $\lambda$ is the order type of the real line. The site labels the problem OPEN, and its only remark credits Erdős and Rado with $\mathfrak c\to(\omega+n,4)^3_2$ for every finite $n\ge2$. That result is Theorem 31, relation (30), of [ErRa56], and it is the accepted partial claim Erdős and Rado 1956, refereed in the Bulletin of the American Mathematical Society; it gives $\lambda\to(\beta,4)^3_2$ for every $\beta<\omega 2$, and so every instance with $\beta<\omega 2$ and $n\le4$. Two families of instances are trivial. For $n\le3$ and every countable $\beta$, a set of three reals all of whose triples are blue is a single blue triple, so either some triple is blue or every triple is red, and in the second case any set of reals of order type $\beta$ is red-monochromatic. For $\beta\le\omega$ and every $n$, Ramsey's theorem applied to the triples of an increasing $\omega$-sequence of reals gives an infinite homogeneous subset, which is either red, of order type $\omega$ and so containing a set of order type $\beta$, or blue and so containing $n$ points. The remaining instances are open: $\beta\ge\omega 2$ with $n\ge4$, of which $(\omega 2,4)$ is the case the formal-conjectures file marks as the first open one beyond Erdős and Rado, and $\omega<\beta<\omega 2$ with $n\ge5$; Erdős [Er87] writes that he knows nothing about replacing $\omega+n$ by a larger countable ordinal or $4$ by a larger $n$. The formal-conjectures file's variant omega_three, the one variant that carries a formal proof, concerns the initial ordinal of the continuum and the trivial instance $(\omega,3)$, so it gets no claim page; its variant erdos_rado states the accepted result with its proof left as sorry. Proof coverage: none of the proofs is reconstructed in this corpus.

Search scope, 2026-10-07: the site's problem page (last edited 23 January 2026) and its discussion thread (no comments and no proof claims), [Er87], [ErRa56] and the formal-conjectures statement file at the commit linked under Formulation; no wider literature search is recorded.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1956_partition_calculus_set_theory
• erdos_1987_problems_finite_infinite_graphs
• erdos_1987_problems_finite_infinite_graphs / problem_3