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Problem 1199
Is it true that in any -colouring of there exists an infinite set such that all elements of are the same colour?
Source: erdosproblems.com/1199
- Additive combinatorics
- Ramsey theory
Status
Claimed, ProvedA full solution has been claimed but not yet accepted. The statement is true.
Open. For three or more colors the answer is no: Hindman [Hi79] (J.
Combin. Theory Ser. A 27 (1979), 19--32, refereed) constructed a three-cell
partition with no infinite having monochromatic
(Theorem 2.4 (Hindman 1979),
p. 21, with its proof on pp. 21--23, followed below); the result rests on the
paper, to which the site, [LeWi24], [HLSXXZ26] and Erdős's 1977 and 1980 surveys
all attribute it; a different explicit three-coloring recorded in an external
Lean file behind the formal-conjectures variant is checked below by an
elementary argument. For two colors the paper itself settles only the admissible
partitions, those with a class containing, for a fixed difference ,
arbitrarily long arithmetic progressions that start at an
even integer
(Corollary 2.10 (Hindman 1979),
p. 27), recorded as an accepted partial claim on
its claim page (Hindman, 1979), and
closes its § 2 with the conjecture that the restriction can be removed (p. 28).
The three-color Theorem 2.4, which the site credits, answers a variant and
settles no instance of the two-color question, so it has no claim page. For two
colors in general, a preprint of Huang, Lian, Shao, Xiao, Xu and Zhang
([HLSXXZ26], arXiv:2607.17333, v3 of 29 July 2026) claims an affirmative answer
(its Theorem 1.1). No refereed version, independent review or citing paper of
that claim was found, and the site had not adopted it (its label was OPEN on
2026-09-18, seven weeks after a thread comment of 1 August 2026 reported it);
the claim is recorded as a pending full claim on its
claim page (Huang, Lian, Shao, Xiao, Xu and Zhang, 2026),
from which the frontmatter standing claimed, proved is derived. For the
two-color statement the absence of acceptance evidence is a bounded negative
finding of the search, not a certificate of openness.