Problem 475

Let $p$ be a prime. Given any finite set $A\subseteq \mathbb{F}_p\backslash \{0\}$ , is there always a rearrangement $A=\{a_1,\ldots,a_t\}$ such that all partial sums $\sum_{1\leq k\leq m}a_{k}$ are distinct, for all $1\leq m\leq t$ ?

Source: erdosproblems.com/475

Number theory
Additive combinatorics

Wiki page

Status

Open

No claim settles this problem.

Decidable, the site's label; the label describes the shape of what remains and is not a theorem, and the question for every prime is open. Proved for all sufficiently large primes $p$ : the site's chain of four range results covers every size $t$ once $p\ge p_0$ , with $p_0$ unspecified in every link, recorded as a pending partial claim on its claim page (Pham Sauermann, 2026) (the site's label leaves the problem open, so the curator's commentary is not acceptance). Small $t$ : Bedert and Kravitz (Israel J. Math. 273 (2026), refereed; claim page (Bedert and Kravitz, 2024), accepted) for $t\le e^{c(\log p)^{1/4}}$ , every $c>0$ and $p$ large, improved to $t\le e^{c(\log p)^{1/3}}$ by Costa and Della Fiore (a 2026 preprint), after Kravitz's $t\le\log p/\log\log p$ for every prime (claim page (Kravitz, 2024), pending). Medium $t$ : Pham and Sauermann (a 2026 preprint) for $C_\alpha\le t\le p^{1-\alpha}$ , any fixed $0<\alpha<1$ . Large $t$ : Bedert, Bucić, Kravitz, Montgomery and Müyesser (a 2025 preprint) for $t\ge p^{1-c}$ in every finite group. Very large $t$ : their Theorem 7.1 (Bedert et al. 2025), $t\ge p-p^{1-\gamma}$ , derived from the random Hall--Paige theorem of Müyesser and Pokrovskiy (Invent. Math. 240 (2025), refereed). For every prime the statement holds for $t\le12$ (Costa and Pellegrini, Arch. Math. 115 (2020), refereed; claim page (Costa and Pellegrini, 2020), accepted), for every set of size $p-2$ or $p-1$ (Bode and Harborth, Discrete Math. 299 (2005), refereed; claim page (Bode and Harborth, 2005), accepted; $t=p-1$ is also Graham's case) and for every $(p-3)$ -subset with nonzero sum (Hicks, Ollis and Schmitt, J. Combin. Des. 27 (2019), refereed; claim page (Hicks, Ollis and Schmitt, 2018), accepted). The site's range $p-3\le t\le p-1$ also includes the zero-sum $(p-3)$ -subsets, which no cited source covers; Kravitz states the range as nonzero-sum sets of size $p-2$ or $p-3$ . Bedert and Kravitz's refereed theorem has its own page. Müyesser and Pokrovskiy do not state the subset case, which Bedert, Bucić, Kravitz, Montgomery and Müyesser make explicit. That paper and Costa and Della Fiore's are preprints. These links are recorded on the chain's page. Two of the four range results are unrefereed preprints, so the completion for large $p$ carries the preprint qualification; none of the sources cited here bounds the finite set of primes left unchecked. Whether the label should stand for a statement proved for all $p\ge p_0$ with $p_0$ unknown is a question about the catalog's vocabulary that this page records and does not decide.

Er73: Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117--138; Section 7, printed pp. 126--127. Library home: Erdos 1973.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980); printed p. 95. Library home: Erdős and Graham 1980.
Gr71: Graham, R. L., On sums of integers taken from a fixed sequence. Proceedings of the Washington State University Conference on Number Theory (1971), 22--40; Question 10, printed p. 36; the monograph's [Gr (71)] and the paper cited by Pham and Sauermann for the conjecture ("[7, p. 36]"). Library home: Graham 1971.
Kr24: Kravitz, N., Rearranging small sets for distinct partial sums. arXiv:2407.01835v2 (18 August 2024), 4 pp. Theorems 1.2 and 1.3, p. 1. Library home: Kravitz 2024.
BeKr24: Bedert, B. and Kravitz, N., Graham's rearrangement conjecture beyond the rectification barrier. arXiv:2409.07403v2 (7 January 2025, "Incorporates referee's suggestions"), 18 pp.; Israel J. Math. 273 (2026), no. 1, 471--500, DOI 10.1007/s11856-025-2871-6 (published online 30 November 2025; Crossref record; not compared). Theorem 1.2, p. 1. Library home: Bedert and Kravitz 2024.
CoDe26: Costa, S. and Della Fiore, S., New bounds for (weak) sequenceability in $\mathbb Z_k$ . arXiv:2602.19989v1 (23 February 2026), 9 pp. Theorem 1.3, p. 2. Library home: Costa 2026.
PhSa26: Pham, H. T. and Sauermann, L., On Graham's rearrangement conjecture. arXiv:2602.15797v1 (17 February 2026), 27 pp. Theorem 1.2, p. 1; Theorem 1.3 and Corollary 1.4, p. 2. Library home: Pham and Sauermann 2026.
BBKMM25: Bedert, B., Bucić, M., Kravitz, N., Montgomery, R. and Müyesser, A., On Graham's rearrangement conjecture over $\mathbb F_2^n$ . arXiv:2508.18254v1 (25 August 2025), 43 pp. Theorem 1.4, p. 3; Theorem 7.1, p. 25; Theorem A.2, p. 42. Library home: Bedert et al. 2025.
MuPo25: Müyesser, A. and Pokrovskiy, A., A random Hall--Paige conjecture. arXiv:2204.09666v3 (25 February 2025, "final version, to appear in Inventiones Mathematicae"), 73 pp.; Invent. Math. 240 (2025), no. 3, 779--867, DOI 10.1007/s00222-025-01328-x (published online 5 March 2025; Crossref record; not compared). Theorem 1.1, p. 3; Theorem 6.9, p. 51. Library home: Müyesser and Pokrovskiy 2022.
CoPe20: Costa, S. and Pellegrini, M. A., Some new results about a conjecture by Brian Alspach. arXiv:2003.05939v2 (23 April 2020), 9 pp.; Arch. Math. (Basel) 115 (2020), no. 5, 479--488, DOI 10.1007/s00013-020-01507-7 (published online 29 August 2020; Crossref record; not compared). Conjecture 1.2, p. 2; Proposition 4.2, p. 6 (arXiv pagination). Library home: Costa and Pellegrini 2020.
HOS19: Hicks, J., Ollis, M. A. and Schmitt, J. R., Distinct partial sums in cyclic groups: polynomial method and constructive approaches. arXiv:1809.02684v1 (7 September 2018), 18 pp.; J. Combin. Des. 27 (2019), no. 6, 369--385, DOI 10.1002/jcd.21652 (published online 31 January 2019; Crossref record; not compared). Conjecture 1.1, p. 2; Theorem 2.2, p. 6; Theorem 4.3, p. 12; Theorem 4.6, p. 15 (arXiv pagination). Library home: Hicks, Ollis and Schmitt 2019.
BoHa05: Bode, J.-P. and Harborth, H., Directed paths of diagonals within polygons. Discrete Math. 299 (2005), 3--10, DOI 10.1016/j.disc.2005.05.006 (Kravitz's [3]; HOS19's [9], the source of the sizes $p-1$ , $p-2$ for Alspach's conjecture). Conjecture 1, printed p. 3; Theorems 1 and 2 with the odd- $n$ half of the proof of Theorem 2, printed p. 4; the even- $n$ induction, pp. 5--9, read for structure only. Library home: Bode and Harborth 2005; result pages Theorem 1 and Theorem 2.
ADMS16: Archdeacon, D. S., Dinitz, J. H., Mattern, A. and Stinson, D. R., On partial sums in cyclic groups. J. Combin. Math. Combin. Comput. 98 (2016), 327--342 (HOS19's [8]; CoPe20's [6]): the paper proving that Alspach's conjecture implies the distinct-partial-sums conjecture (its Proposition 1.1; arXiv:1501.06872), for sets of size at most $k$ in the same group as [CoPe20], p. 2, states it, while [HOS19], p. 2, gives no sizes; a zero-sum set of size $t$ needs an Alspach ordering of one of its subsets of size $t-1$ ([CoPe20], p. 7). Not held; quoted from [HOS19], p. 2, and [CoPe20], p. 2.
BFMPY25: Bucić, M., Frederickson, B., Müyesser, A., Pokrovskiy, A. and Yepremyan, L., Towards Graham's rearrangement conjecture via rainbow paths. arXiv:2503.01825 (2025); the approximate version (all but $o(|S|)$ partial sums distinct) as described in [PhSa26], p. 1, and [BBKMM25], p. 2. Not held; title only.
CDFL26: Costa, S., Della Fiore, S., Feng, T. and Liu, H., Kneserized anticoncentration and reverse absorption for Graham's rearrangement conjecture. arXiv:2608.10015v2 (18 August 2026); composite cyclic groups $\mathbb Z_{tp}$ and products of large primes. Abstract only (arXiv API); it is context for the problem, not a result about it.

Problem 475

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Proof claims

Discussion