Hindman 2017 pairwise sums colourings reals

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corollary_4_5: For any countable coloring of the reals whose color classes all have the property of Baire, or are all Lebesgue measurable, and any k at least two, some set H of size continuum has its k-fold sumset kH monochromatic; so a coloring refuting Problem 965 cannot have regular color classes.

theorem_2_8: For each finite n the group G(omega_n), a direct sum of omega_n copies of the rationals, has a coloring in 2^{4+n}·3^2 colors under which no infinite set has its sums of k distinct elements together with its k-multiples monochromatic, so in particular no infinite X has kX monochromatic.

theorem_3_2: A ZFC two-coloring of the reals, built from a Hamel basis and a well-ordering of order type continuum, under which no set of size continuum has its sums of k distinct elements monochromatic, for any k at least two; under the continuum hypothesis this refutes Problem 965.

theorem_4_4: For any k at least two and any subset Z of (0,1) that is non-meagre with the property of Baire or Lebesgue measurable of positive measure, some set H of reals of size continuum has its k-fold sumset kH contained in Z.

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Hindman, Neil and Leader, Imre and Strauss, Dona, Pairwise sums in colourings of the reals. Abh. Math. Semin. Univ. Hambg. 87 (2017), no. 2, 275--287, doi:10.1007/s12188-016-0166-x (published online 21 December 2016; the publisher's record read). The copy read for this card is arXiv:1505.02500v1 (11 May 2015; the only arXiv version listed on 2026-09-18), 17 pages with its own pagination and a complete text layer; every locator below is a page of that preprint, and the journal text was not compared. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1505.02500), every other right reserved.

The paper asks which sumset structures must appear inside a color class when the reals are finitely colored. Theorem 3.2 (p. 9) gives a 2-coloring of R such that for every k > 1 no set X of size continuum has FS_k(X) monochromatic, built from a Hamel basis together with a well-ordering of R of order type c. It is a ZFC theorem about sets of size c, and its bearing on sets of size aleph_1 needs CH: the introduction (p. 2) says that the proof "relies on the Continuum Hypothesis (CH)", that the authors "do not know whether or not CH is needed" and that "Without CH, our result asserts that there is no such set of size c", and the remark after the proof (p. 11) shows that when c > omega_1 the coloring of Theorem 3.2 does have a set X of size omega_1 with FS_k(X) monochromatic for every k > 1; Question 3.3 (p. 11) asks for a ZFC finite coloring of R with no uncountable X having FS_2(X) monochromatic. Theorem 1.2 (p. 4) gives, for k < m, a finite coloring with Borel classes blocking FS_k(X) union FS_m(X) on every uncountable set. Theorem 2.8 (p. 8) colors, for each n < omega, the group G(omega_n) with 2^{4+n}·3^2 colors so that no infinite X has FS_k union {kx : x in X} monochromatic (the statement is printed with "infinite subset X of G(omega_1)", a misprint for G(omega_n)), and "In particular, there is no infinite subset X of G(omega_n) for which kX is monochromatic"; since R is isomorphic to G(c), this gives under CH, and more generally when c < aleph_omega (p. 2; the paragraph before Theorem 2.8 says the smallest value of c at which the assertion might fail is omega_{omega+1}), a finite coloring of R with no infinite X having FS_k union {kx} monochromatic, hence none with kX monochromatic; Question 2.9 (p. 8) asks whether this is provable in ZFC alone. In the positive direction Theorem 4.4 (p. 13) shows that if Z is a subset of (0,1) that is either non-meagre with the Baire property or of positive Lebesgue measure, then there is H of size continuum with kH contained in Z, so Corollary 4.5 (p. 13) gives a monochromatic kH of size continuum for any countable coloring whose classes are all Baire or all measurable (Theorem 4.2 and Corollary 4.3, p. 12, give the uncountable version in the Baire case by a simpler argument). Methods are a stepping-up argument over the groups G(kappa) plus measure/category regularity arguments. For problem 965 the paper is the source of the negative answer under CH (Theorem 3.2 with k = 2 and c = aleph_1), of the explicit statement that its coloring does not settle the question in ZFC, and of the positive result for Baire or measurable colorings; the ZFC negative answer is Komjáth's and Soukup and Weiss's, recorded on the problem page.

Read status: claims checked for Definition 1.1, Theorem 3.2, the remark after its proof, Question 3.3, Theorem 4.4 and Corollary 4.5, read clause by clause in the text layer and, for pp. 9 and 11, on the rendered pages on 2026-09-18; Theorem 1.2, Theorem 2.8, Definition 2.6, Lemma 2.7, Question 2.9, Theorem 4.2 and Corollary 4.3 were read clause by clause on the rendered pages; the proofs of Theorem 3.2 (pp. 9--11), of Theorem 2.5 and Lemma 2.7 (pp. 6--8) and of Theorem 4.4 (pp. 13--16) were read for their structure and not checked step by step. Result pages: Theorem 2.8, Theorem 3.2, Theorem 4.4, Corollary 4.5.

Source: https://arxiv.org/abs/1505.02500.

Bears on. #965: Theorem 3.2 (p. 9) with k = 2 is the negative answer under CH; the introduction's sentence on CH (p. 2), the remark after the proof and Question 3.3 (p. 11) record that the coloring does not settle the question in ZFC; Corollary 4.5 (p. 13), deduced from Theorem 4.4 (p. 13), gives with k = 2 a set of size continuum whose sums of two distinct elements share a color, for every 2-coloring whose classes have the property of Baire or are measurable.

Results to transcribe.

• Theorem 1.2 (p. 4): For k < m there is a finite coloring of R with no uncountable X having FS_k(X) union FS_m(X) monochromatic; its color classes are Borel.
• Theorem 2.8 (p. 8): For n < omega, G(omega_n) can be colored with 2^{4+n}·3^2 colors so that no infinite X has FS_k union {kx : x in X} monochromatic, and in particular no infinite X has kX monochromatic (the printed statement says "G(omega_1)" where the argument and the introduction have G(omega_n)); under CH, or under c < aleph_omega, this gives a finite coloring of R with the same property.
• Theorem 3.2 (p. 9): There is a 2-coloring of R such that for every k > 1 no X of size continuum has FS_k(X) monochromatic.
• Remark after Theorem 3.2 (p. 11): If c > omega_1 there is a set X of size omega_1 such that FS_k(X) is monochromatic for every k > 1 under the coloring psi of Theorem 3.2 (X = {e_i + e_j : i W j} for a j with omega_1 elements before it in W).
• Question 3.3 (p. 11): Can one show in ZFC that there is a finite coloring of R such that no uncountable X has FS_2(X) monochromatic?
• Theorem 4.2 and Corollary 4.3 (p. 12): If Z is nonmeagre with the property of Baire there is an uncountable H with kH contained in Z; so a countable coloring with Baire classes has an uncountable H with kH monochromatic.
• Theorem 4.4 (p. 13): If Z is a subset of (0,1) that is non-meagre with the Baire property or has positive measure, there is H of size continuum with kH contained in Z.
• Corollary 4.5 (p. 13): For any countable coloring of R whose classes are all Baire or all measurable, there is H of size continuum with kH monochromatic.
• Question 2.9 (p. 8): Asks whether ZFC alone gives a finite coloring of R with no infinite X having FS_k union {kx} monochromatic.

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