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The claim. There is a constant $C$ such that for every prime $p>C$ and every $p$ nonzero residues $a_1,\ldots,a_p$ modulo $p$ with a single value of $k$ for which some $k$ of them sum to $0$ modulo $p$, at most two distinct residues occur among the $a_i$. The source is P. Erdős and E. Szemerédi, On a problem of Graham, Publ. Math. Debrecen 23 (1976), no. 1--2, 123--127, DOI 10.5486/pmd.1976.23.1-2.20 (the byline prints "E. Erdős" [sic], a misprint for P. Erdős; the paper is item 1976-18 of Erdős's publication list), paged as the main theorem of Erdős and Szemerédi (1976). Theorem 1 (p. 123) is the sharper tool: for $\eta<\eta_0$ small enough, $p>p_0(\eta)$ and a set $A$ of $l>\eta^{1/10}p$ nonzero residues in which no residue occurs $\eta p$ times or more, every residue is a nonempty $0$--$1$ combination of the elements of $A$. When every residue has multiplicity below $\eta_0p$, splitting $a_1,\ldots,a_p$ into two sets satisfying the hypothesis shows that zero sums of two different lengths exist; the case of a residue of high multiplicity occupies pp. 125--127. The authors write that extending the proof to small $p$ would need heavy computation but no new idea, and that their proof is surprisingly complicated, though they are not convinced that no simpler proof is possible. The conjecture, Theorem 1 and the deduction paragraph (p. 123) were checked; the proof (pp. 123--127) was not read.

Covers. The statement of Problem 541 for all sufficiently large primes $p$, with the $a_i$ nonzero residues. Not covered: the small primes, which the paper leaves to computation, and sequences containing the residue $0$, which the site's wording admits and the paper's statement excludes. Both are covered by the accepted full claim Gao, Hamidoune and Wang 2009, and for every finite abelian group by Grynkiewicz 2009.

Acceptance. Refereed: the journal publication cited above. Reviewed: the site's curator, Thomas Bloom, who is independent of the authors, credits the large-prime case to this paper in the problem's commentary (page last edited 8 April 2026); Erdős's 1980 survey (p. 112) and the Erdős--Graham monograph of 1980 (p. 95) record the theorem as proved, the monograph calling the proof unexpectedly complicated; Gao, Hamidoune and Wang (2010) and Grynkiewicz (2011) each restate it as Graham's conjecture for large primes before extending it. Nothing here is independently reviewed by this project.

Date. The paper prints "Received February 14, 1974" on its last page (p. 127); the page name uses that received date, the paper's first dated record.

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