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Claim. There is a $2$-coloring of $\mathbb R$ such that for every $k\ge2$ no set $X\subseteq\mathbb R$ with $|X|=\mathfrak c$ has $FS_k(X)$, the set of sums of $k$ distinct elements of $X$, monochromatic. The coloring fixes a Hamel basis $\langle e_i\rangle_{i\in\mathbb R}$ of $\mathbb R$ over $\mathbb Q$ and a well-ordering $W$ of $\mathbb R$ of order type $\mathfrak c$, and colors $x$ by the parity of the position, among the support indices of $x$ in increasing order, of the $W$-largest one. This is Theorem 3.2 of the paper, paged at Theorem 3.2 of the library's source card. With $k=2$ it says that no set of size $\mathfrak c$ has the sums of its distinct pairs in one color.

Hypothesis. The theorem itself is a theorem of ZFC, but it concerns sets of size $\mathfrak c$, while Problem 965 asks about sets of size $\aleph_1$. Under the continuum hypothesis $\aleph_1=\mathfrak c$, and the case $k=2$ is the negative answer to the problem; this is how the paper presents the result, saying that its proof relies on CH and that without CH it asserts only that there is no such set of size $\mathfrak c$. The authors add (p. 11 of the preprint) that if $\mathfrak c>\omega_1$ their coloring does admit a set of size $\omega_1$ with $FS_k$ monochromatic for every $k\ge2$, and ask (Question 3.3) whether ZFC alone gives a finite coloring with no uncountable set whose pair sums are monochromatic. That question is answered by the accepted ZFC claims Komjáth 2016 and Soukup and Weiss 2015, which reach the conclusion without CH; CH itself is independent of ZFC, and this claim remains conditional on it.

Acceptance. Refereed: N. Hindman, I. Leader and D. Strauss, Pairwise sums in colourings of the reals, Abh. Math. Semin. Univ. Hambg. 87 (2017), no. 2, 275--287, published online 21 December 2016 (the publisher's record), following the preprint arXiv:1505.02500 of 11 May 2015, the date this page is named by. Reviewed: the site's curator, Thomas Bloom, credits the paper as the published proof of the disproof under the continuum hypothesis, in the stronger $k$-fold form, in the problem's commentary (page last edited 16 January 2026, accessed 2026-09-18). Semantic Scholar's four citing records, include by title no dispute.

Read depth. The basis is arXiv v1, the only arXiv version; the journal text was not compared. Theorem 3.2 (p. 9), the remark and Question 3.3 (p. 11) were checked clause by clause; the proof (pp. 9--11) was read for its structure only, and nothing is independently reviewed in this corpus.