---

The claim. Let $G$ be a finite abelian group of order $n$ and $S$ a sequence of $n$ elements of $G$. If there is exactly one $r\in[1,n]$ for which some $r$ terms of $S$ sum to $0$, then $S$ has at most two distinct terms; the theorem also gives explicit forms that every such sequence must take, five for cyclic $G$, stated as necessary conditions only. With $G=\mathbb Z/p$ this is the statement of Problem 541 for every prime $p$, the residue $0$ admitted, and with $G=\mathbb Z/n$ the statement for every modulus. The source is D. J. Grynkiewicz, Note on a conjecture of Graham, European J. Combin. 32 (2011), no. 8, 1336--1344, DOI 10.1016/j.ejc.2011.06.004, arXiv:0903.3200v1 (18 March 2009, the only arXiv version and the date this page is named by), carded as Grynkiewicz (2011); Theorem 3.4 is on p. 5 of the preprint, and the journal text was not compared. The introduction presents the result as a short proof of the original conjecture using only the Cauchy--Davenport theorem and the pigeonhole principle, every non-prime order (composite cyclic moduli and non-cyclic groups) going through the DeVos--Goddyn--Mohar theorem instead; the proof (pp. 6--10) was not read here, and the statement was checked clause by clause.

Acceptance. Refereed: the journal publication cited above; the issue is dated November 2011 in the Crossref record (2026-09-18). The site's problem page does not cite the paper; a thread comment of 14 April 2026 cites it as [Gr11] and points to it as an easy proof of the Erdős--Szemerédi theorem. The commentary of the site's curator, Thomas Bloom, credits the problem's proof to Gao, Hamidoune and Wang 2009 and does not name this paper, so the site's acceptance is not listed as reviewed here. The header of the Lean proof recorded as the accepted Lean claim says that its argument is closer to this paper's than to the two it cites. Nothing here is independently reviewed by this project.

Depends on. Nothing in this wiki: the theorem is proved within the paper, whose card is linked above.