Problem 172

Is it true that in any finite colouring of $\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour?

Source: erdosproblems.com/172

Additive combinatorics
Ramsey theory

Wiki page

Status

ClaimedProved

A full solution has been claimed but not yet accepted. The statement is true.

OPEN, the site's label (page last edited 6 April 2026). No proof or disproof of the statement over $\mathbb{N}$ was found in the search whose scope the Current assessment records. The strongest results are the case $n=2$ for two colors (by computer in Hindman 1979 and without a computer in Bowen 2022, both refereed), the pattern $\{x,x+y,xy\}$ for all finite colorings (Moreira 2017, refereed), and the full statement over $\mathbb{Q}$ (Bowen and Sabok for $n=2$ , refereed; Alweiss for all $n$ , a preprint accepted per its arXiv comment). Even the case $n=2$ over $\mathbb{N}$ with three or more colors was open at that date. This is a bounded negative finding, not a certificate of openness. After that search, the OpenAI release's preprint of 23 September 2026 claimed the full statement for every $n$ and every finite coloring; it is recorded on the claim page OpenAI 2026 as claimed, since it is unrefereed, unreviewed and has no Lean statement of the theorem, and the derived standing is claimed through that page.

Er77c: Erdős, P., Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976), Lecture Notes in Math. 626, Springer (1977), 43--72; p. 58. Library home: Erdos 1977.
ErGr79: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory: van der Waerden's theorem and related topics. Enseign. Math. (2) 25 (1979), 325--344; pp. 329--330. Library home: Erdős and Graham 1979.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980). Library home: Erdős and Graham 1980; its passage on this problem is on printed pp. 13--14: citing [Hi (79) b] and [Hi (80)], it gives the 1979 chapter's account of Hindman's two- and seven-class partitions in which no infinite set has all its pair sums and all its finite (two classes) or pair (seven classes) products in one class, and closes with the same sentence, "Whether arbitrarily large finite sets $\{x_1,\ldots,x_k\}$ with this property can always be found for any partition of $\mathbf{N}$ into finitely many classes is completely open."
Hi80: Hindman, N., Partitions and sums and products---two counterexamples. J. Combin. Theory Ser. A 29 (1980), no. 1, 113--120, doi:10.1016/0097-3165(80)90052-7. Theorem 2.14 (p. 117) and Theorem 2.15 (p. 118) refute the infinite version with two and with seven colors, and Question 3.3 (p. 120) states this problem as open. Library home: Hindman 1980 and its Hindman 1980, Theorem 2.14, Hindman 1980, Theorem 2.15 and Hindman 1980, Question 3.3 pages.
Mo17: Moreira, J., Monochromatic sums and products in $\mathbb{N}$ . Ann. of Math. (2) 185 (2017), no. 3, 1069--1090, doi:10.4007/annals.2017.185.3.10; arXiv:1605.01469v1 (5 May 2016). Library home: Moreira 2017.
BoSa22: Bowen, M. and Sabok, M., Monochromatic products and sums in the rationals. arXiv:2210.12290v1 (21 October 2022, the version read; no file held); Forum Math. Pi 12 (2024), e17, doi:10.1017/fmp.2024.19. Library home: Bowen and Sabok 2022.
Al23: Alweiss, R., Monochromatic sums and products over $\mathbb{Q}$ (the site's reference entry describes it as the rational case of Hindman's conjecture). arXiv:2307.08901, v1 18 July 2023 to v6 12 July 2026; the v6 arXiv comment reads "accepted in Duke Math Journal"; no journal record found. Library home: Alweiss 2023.
Bo22: Bowen, M., Monochromatic products and sums in 2-colorings of $\mathbb{N}$ . arXiv:2205.12921v1 (25 May 2022); Adv. Math. 462 (2025), 110095, doi:10.1016/j.aim.2024.110095. Library home: Bowen 2022.
HIL23: Hindman, N., Ivan, M.-R. and Leader, I., Some new results on monochromatic sums and products in the rationals. New York J. Math. 29 (2023), 301--322; arXiv:2210.07831v3. No library home; cited from its abstract only.
ABS25: Alweiss, R., Bowen, M. and Sabok, M., Sums, products, and exponents in two-colorings of the naturals. arXiv:2512.09598v1 (10 December 2025); preprint, cited from its abstract only.
GrSa25: Green, B. and Sawhney, M., Bounds for monochromatic solutions to $\{x+y,xy\}$ . arXiv:2511.09365v2 (19 November 2025); preprint, cited from its abstract only.

Problem 172

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