Leader 2024 monochromatic sumsets countable colourings abelian groups

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Imre Leader, Kada Williams, Monochromatic Sumsets in Countable Colourings of Abelian Groups. arXiv:2407.03938 [math.CO], v1 posted 4 July 2024 (the title page is dated July 8, 2024); the retained folder-name PDF is that version, six pages with a complete text layer. On 2026-09-18 the arXiv record showed no later version and no journal reference or DOI, so no published version is identified here. The arXiv record (https://arxiv.org/abs/2407.03938, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Fernandez-Breton, Sarmiento and Vera showed that countably coloring a large direct sum of copies of Z_4 yields arbitrarily large finite sets X with X+X monochromatic, and asked whether elements of order 4 are necessary in a strong sense. Leader and Williams answer yes: by Theorem 1 (p. 2), every abelian group G with no element of order 4 has a countable coloring under which no two distinct elements x and y have 2x, 2y and x+y all of one color, so there is not even a monochromatic sumset X+X with |X| = 2. The introduction (p. 1) opens with the naturals: by Ramsey's theorem any finite coloring has an infinite X with all pairwise sums of distinct elements one color, while for the full sumset X+X (including the terms x+x) "it is a simple matter to find a 3-colouring yielding no such monochromatic set", and then: "It is worth mentioning that, surprisingly, it is unknown as to whether or not this can be achieved with a 2-colouring: this is called Owings' problem [15]. For background on this, see [8]." Reference [15] is J. Owings, Problem E2494, Amer. Math. Monthly 81 (1974), 902, and [8] is Hindman, Leader and Strauss, "Pairwise sums in colourings of the reals", Abh. Math. Sem. Univ. Hamburg 87 (2017), 275-287. The introduction goes on (pp. 1-2) to the rationals (a 12-coloring of Q with no infinite monochromatic sumset), the reals under CH (Hindman, Leader and Strauss), the positive results of Leader and Russell in large rational dimension, of Komjath, Leader, Russell, Shelah, Soukup and Vidnyanszky for the reals under a large-cardinal assumption, and Zhang's proof without that assumption that the positive answer for the reals is consistent, and to countable colorings of finite subsets; these are reported as the introduction states them and were not checked against their sources.

Read status: claims checked for the two Owings sentences and the 3-coloring remark of the introduction (p. 1, read on the rendered page image and in the text layer) and for the statement of Theorem 1 (p. 2, read on the rendered page image); the proof of Theorem 1 (pp. 3-5) was not read. For problem 1199 the paper is a dated attestation (July 2024) that the 2-coloring case of Owings' problem was regarded as open; Theorem 1 itself concerns countably many colors on general abelian groups and neither proves nor disproves the 2-coloring statement over the naturals.

Source: https://arxiv.org/abs/2407.03938.

Bears on. #1199: the introduction's sentence that the 2-coloring case "is unknown" (July 2024) and its reference to Hindman, Leader and Strauss for background; Theorem 1 is context only.

Results to transcribe.

• Theorem 1 (p. 2): "Let G be an abelian group not containing any elements of order 4. Then there is a countable colouring of G for which there do not exist distinct x and y with {2x, 2y, x + y} monochromatic."
• Introduction (p. 1): for finite colorings of the naturals, an infinite X with all pairwise sums of distinct elements monochromatic exists by Ramsey's theorem; a 3-coloring with no infinite X having X+X monochromatic is "a simple matter"; whether a 2-coloring can do this "is unknown" (Owings' problem, reference [15]).