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Claim. There is a coloring $F:\mathbb R\to2$ such that for every uncountable $X\subseteq\mathbb R$ and every $N\ge2$ the sums $\sum E$ over $E\in[X]^N$, the $N$-element subsets of $X$, take both colors. With $N=2$, every uncountable $X$, so every $X$ of cardinality $\aleph_1$, has sums of two distinct elements in both colors, and the answer to Problem 965 is no in ZFC. This is Corollary 3.2 (p. 4) of the manuscript, paged at Corollary 3.2 of the library's source card. It follows from Theorem 3.1 (p. 3), which colors the finite subsets of $2^\omega$ so that every uncountable family and every $N\ge2$ have $N$ distinct members whose union has either color, by the Sierpiński coloring applied to the pair realizing the maximal splitting level of a finite set; Lemma 2.1 transfers the union form to sums of reals through a Hamel basis. The manuscript adds that under CH the coloring can use $2^\omega$ colors (Corollary 2.2) and that three colors cannot be guaranteed in ZFC, by a consistency result of Shelah (Corollary 2.3). It records (p. 1) that Komjáth proved the same result independently; Komjáth's refereed paper is the accepted claim Komjáth 2016.

Acceptance. Reviewed: the site's curator, Thomas Bloom, adopted the manuscript as one of two independent ZFC disproofs in the problem's commentary and relabeled the problem DISPROVED (page last edited 16 January 2026, accessed 2026-09-18), after the thread's comment of 2 January 2026 pointed to it; the formal-conjectures statement file for the problem cites it beside Komjáth's paper in its docstring, from a second copy on the first author's site (the github.io link above), which was not compared with the renyi.hu copy. Not refereed: the manuscript is unpublished, with no journal or arXiv record found on 2026-09-18 (Crossref bibliographic query, arXiv author and abstract queries, Semantic Scholar), and no independent review of it was found. The acceptance rests on the curator's adoption; Komjáth's refereed publication of the same theorem is the problem's other accepted claim, not evidence listed here.

Postings. The copy on the first author's github.io site (the second link) is the one the formal-conjectures docstring cites and the one that answered on 2026-10-07. The renyi.hu address (the first link) is the original location, which answered on 2026-09-05; it returned HTTP 404 on 2026-10-02 and is listed as the original posting.

Dating. The manuscript's text carries no date; its PDF metadata gives a creation date of 7 September 2015, a typesetting date and not a posting date, so this page is named by the month and the day is a placeholder.

Read depth. Claims checked for Theorem 3.1 and Corollary 3.2 (pp. 1, 3 and 4), with pp. 1--5 read; the two-page proof was read for structure only, and nothing is independently reviewed in this corpus.