Problem 475

../

claims/: The 6 claim pages of Problem 475, one per claimant's result; the problem's standing derives from them.

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Statement. 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 $1\leq m\leq t$?

Formulation. The site's wording (page last edited 5 March 2026). An ordering with this property is called a valid ordering in the papers; only the partial sums with $m\ge1$ are compared, so a valid ordering may end at $0$ when the elements of $A$ sum to $0$. Alspach's conjecture, which the site mentions, is the stronger statement for cyclic groups $\mathbb Z_n$ that asks the partial sums to be distinct and nonzero when the total sum is nonzero; the papers call the site's statement over $\mathbb Z_n$ the G-ADMS or distinct-partial-sums conjecture. The origin passages: Erdős's 1973 chapter [Er73], printed pp. 126--127: "Let $a_1,\ldots,a_k$ be $k$ distinct residues mod $p$, $k<p$. Is it true that there is a permutation $a_{i_1},\ldots,a_{i_k}$ so that none of the sums $a_{i_1}+\cdots+a_{i_r}$, $1\le r\le k$ are $\equiv$ (mod $p$)? Graham proved this if $k=p-1$, but the general case is not yet settled" (the print omits the words between "are" and "$\equiv$"; the intended reading is that no two of the sums are congruent); and the 1980 monograph [ErGr80], printed p. 95: "An old question of Graham [Gr (71)] asks if for any set $\{a_1,\ldots,a_t\}$ of nonzero residues modulo a given prime $p$, there is always a rearrangement $(a_{i_1},a_{i_2},\ldots,a_{i_t})$ so that all the partial sums $\sum_{k=1}^ma_{i_k}$ are distinct modulo $p$?" Graham's 1971 paper [Gr71] states the conjecture as Question 10 of its closing section (printed p. 36): "Let $p$ be a prime and suppose $a_1,\ldots,a_k$ are distinct nonzero elements of $Z_p$. Conjecture: There always exists an arrangement $a_{i_1},\ldots,a_{i_k}$ of the $a_i$ such that all partial sums $\sum_{j=1}^ta_{i_j}$, $1\le t\le k$, are distinct modulo $p$"; the paper proves nothing about it and does not mention the case $k=p-1$. The label DECIDABLE is the site's, which the site explains as resolved except for a finite check.

Status. 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 (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, 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, 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, $t\ge p-p^{1-\gamma}$, 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, accepted), for every set of size $p-2$ or $p-1$ (Bode and Harborth, Discrete Math. 299 (2005), refereed; claim page, 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, 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.

Source. erdosproblems.com/475, accessed 2026-09-18: the problem page (DECIDABLE, a label the site explains as resolved except for a finite check; last edited 5 March 2026; source keys [Er73], [ErGr80]; the commentary summarized below; a thanks line naming four contributors), its three-comment discussion thread (23 February, 24 February and 3 March 2026) and its empty proof-claim tab. Cite as: T. F. Bloom, Erdős Problem #475, https://www.erdosproblems.com/475, accessed 2026-09-18.

References.

• [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_problems_results_combinatorial_number_theory.
• [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: erdos_1980_old_new_problems_results_combinatorial_number_theory.
• [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_sums_integers_taken_fixed_sequence.
• [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_rearranging_small_sets_distinct_partial_sums.
• [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_2024_graham_s_rearrangement_conjecture_beyond_rectification.
• [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_new_bounds_weak_sequenceability.
• [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_2026_graham_s_rearrangement_conjecture.
• [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_2025_graham_s_rearrangement_conjecture_over.
• [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: muyesser_2022_random_hall_paige_conjecture.
• [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_2020_new_results_about_conjecture_brian_alspach.
• [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_2019_distinct_partial_sums_cyclic_groups_polynomial.
• [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_harborth_2005_directed_paths_diagonals_within_polygons; 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.

Formalization. None. No file ErdosProblems/475.lean exists in google-deepmind/formal-conjectures at its main-branch commit of 2026-09-18 (the 673-entry directory listing); the site's indicator reads "Formalised statement? No", and the community database (teorth/erdosproblems, as of 2026-09-18) lists the problem as decidable, as of its last update on 23 February 2026, unformalized, with no formal proof.

Current assessment

The question (site formulation as accessed 2026-09-18). The statement above; DECIDABLE, a label the site explains as resolved except for a finite check; last edited 5 March 2026. The commentary, in this page's words: the problem is Graham's, who proved the case $t=p-1$; Alspach made the analogous conjecture for arbitrary abelian groups; the literature calls such an ordering valid; the statement is known for $t\le12$ (Costa and Pellegrini [CoPe20] and their references) and for $p-3\le t\le p-1$ (Hicks, Ollis and Schmitt [HOS19] and their references), a range wider than its sources state (see below); and it is proved for all sufficiently large primes as the consequence of four kinds of result, each covering one range of $|A|$ by its own method: small, Kravitz [Kr24] for $t\le\log p/\log\log p$ (which the site says Will Sawin had observed earlier in a MathOverflow post), Bedert and Kravitz [BeKr24] for $t\le e^{c(\log p)^{1/4}}$, Costa and Della Fiore [CoDe26] for $t\le e^{c(\log p)^{1/3}}$; medium, Pham and Sauermann [PhSa26] for $1\ll_\alpha t\le p^{1-\alpha}$; large, Bedert, Bucić, Kravitz, Montgomery and Müyesser [BBKMM25] for $p^{1-c}\le t\le(1-o(1))p$; very large, Müyesser and Pokrovskiy [MuPo25] for $t\ge(1-o(1))p$. The thread: a comment of 23 February 2026 reporting [MuPo25], [BBKMM25] and [PhSa26]; a comment of 24 February 2026 by an author of [BBKMM25] saying that the solution for large $p$ is spread over four papers, all four needed to cover the whole range, and that the ultra-dense case comes from [MuPo25] but was made explicit only in [BBKMM25]; and a comment of 3 March 2026 reporting [CoDe26]. The proof-claim tab is empty. The community database lists the problem as decidable, as of its last update on 23 February 2026. The reduction to the finite check is a pending partial claim on its claim page: the label leaves the problem open, so the commentary is not acceptance.

The origin. [ErGr80], printed p. 95, and [Er73], printed pp. 126--127, are quoted under Formulation; the 1973 text also states, just before it, a second problem of Graham on $p$ not necessarily distinct residues with a zero-sum condition, and the 1980 text follows the question with a related result of Erdős and Szemerédi. Both name Graham's paper [Gr71], whose Question 10 (printed p. 36) is the conjecture quoted under Formulation, the passage [PhSa26] cites for it ("posed by Graham [7, p. 36] in 1971 and later reiterated by Erdős and Graham [6, p. 95]"); the 1973 text's second problem of Graham is its Question 11 on the same page, and the paper prints no proof of any case, so the case $t=p-1$ rests on the 1973 attribution. The cyclic-group form is Conjecture 1.2 (G-ADMS) of [CoPe20] and Conjecture 1.2 of [HOS19]; Alspach's conjecture is Conjecture 1.1 in both.

Every prime: the finite ranges. $t\le12$: Proposition 4.2 of [CoPe20] (p. 6): "G-ADMS conjecture holds for subsets of size $k\le12$ of cyclic groups of prime order", by Alon's Combinatorial Nullstellensatz applied in the manner of [HOS19] with two computed coefficients whose greatest common divisor is $2^3$, so one is nonzero modulo every odd prime. $p-3\le t\le p-1$: Graham's own case $t=p-1$ (the site; [Er73] p. 127); Theorem 4.6 of [HOS19] (p. 15), Alspach's conjecture for $n=p$ prime and $k=p-3$ by an explicit construction from rotational sequencings and graceful permutations, together with Theorem 2 of [BoHa05] (p. 4), "Conjecture 1 is true for $t=n-2$", Alspach's conjecture for every $n$ and every subset missing one nonzero element (for odd $n$ by an explicit directed cycle through all $n-1$ lengths with one diagonal deleted, for even $n$ by an induction on the missing length carried by the paper's figures), gives Alspach's conjecture for $k\ge p-3$, hence the site's statement for every $(p-2)$-subset and every $(p-3)$-subset with nonzero sum. The appending step in the proof of the Archdeacon--Dinitz--Mattern--Stinson implication (quoted from [HOS19], p. 2, and [CoPe20], pp. 2 and 7; [ADMS16] is not held) gives $\mathbb Z_p\setminus\{0\}$ from the size $p-2$. The zero-sum $(p-3)$-subsets $\mathbb Z_p\setminus\{0,x,-x\}$ would need the size $p-4$ (the proof of [HOS19]'s Theorem 4.6 sets them aside, p. 16), so the site's range goes beyond its sources ([Kr24], p. 1). [BoHa05]'s Theorem 1 (p. 4), "Conjecture 1 is true for $t=n-1$", the other size [HOS19] reports from it, has content only for even $n$: its proof says the only $(n-1)$-subset sums to "$\binom n2$, which is $\not\equiv0\pmod n$ only for $n$ even", so for an odd prime the theorem is vacuous and the case $t=p-1$ remains Graham's. [HOS19]'s Theorem 2.2 (Alspach's conjecture for $k\le10$) is superseded for this problem by [CoPe20]. Acceptance: Arch. Math., Discrete Math. and J. Combin. Des. are refereed journals, the evidence of the three accepted partial claims (Costa and Pellegrini, Bode and Harborth, Hicks, Ollis and Schmitt); the computations behind both theorems were not replayed.

Large primes: the four ranges. The statements are checked clause by clause against the arXiv versions cited; no proof was read beyond its sketch.

• Small. Theorem 1.2 of [Kr24]: for every prime $p$, every $A\subseteq\mathbb F_p\setminus\{0\}$ with $|A|\le\log p/\log\log p$ has a valid ordering, by Lev's rectification of $A\cup\{0\}$ to the integers and the inductive Theorem 1.3 (every finite set of nonzero integers has a valid ordering with the positive elements first); a four-page preprint whose two proofs were read in full (claim page, pending). Theorem 1.2 of [BeKr24]: for every constant $c>0$ and every large prime $p$, every $A$ with $|A|\le e^{c(\log p)^{1/4}}$ has a (two-sided) valid ordering, by a structure theorem into dissociated sets plus a rectifiable residual set and random orderings of the dissociated blocks; refereed (Israel J. Math. 2026; claim page, accepted). Theorem 1.3 of [CoDe26]: there is $c>0$ such that every $A\subseteq\mathbb Z_k\setminus\{0\}$ with $|A|\le\exp(c(\log p)^{1/3})$, $p$ the least prime divisor of $k$, is sequenceable (valid, with nonzero proper partial sums); for $k=p$ this is the site's current small range, with an existential constant where [BeKr24] allows every $c$; a preprint.
• Medium. Theorem 1.2 of [PhSa26]: for any $0<\alpha<1$ there is $C_\alpha$ such that for every prime $p$, every $S\subseteq\mathbb Z_p\setminus\{0\}$ with $C_\alpha\le|S|\le p^{1-\alpha}$ has a valid ordering; the input is the anticoncentration Theorem 1.3, $\max_z\Pr[\Sigma(R)=z]\le1/p+C/(|S|\sqrt m)$ for a uniform $m$-subset $R$ with $C\log|S|\le m\le10^{-3}|S|/\log|S|$, and a random ordering is repaired locally at each zero-sum segment; a preprint, which states (p. 2) that together with the earlier results it "completely settles Graham's rearrangement conjecture for all sufficiently large primes $p$".
• Large. Theorem 1.4 of [BBKMM25]: an absolute $c>0$ such that in every finite group $G$ every $S\subseteq G\setminus\{\mathrm{id}\}$ with $|S|\ge|G|^{1-c}$ has a valid ordering, by the absorption method with a Cayley-graph regularity decomposition (Theorem 1.5); a preprint.
• Very large. Theorem 7.1 of [BBKMM25], labeled as [MuPo25]'s: for $\gamma>0$ and $N$ large, every $S\subseteq G\setminus\{\mathrm{id}\}$ with $|S|\ge N-N^{1-\gamma}$ in a group of order $N$ has a valid ordering; proved as Theorem A.2 from [MuPo25]'s Lemma 6.22 and the method of its Theorem 6.9, which [MuPo25] records for $\gamma\ge1/2$ (a rainbow Hamilton path with prescribed endpoints in the division digraph of a large group, from the random Hall--Paige theorem and sorting networks); [MuPo25] itself does not state the subset case, and the reading of its Theorem 6.9 as the range $t\ge p-p^{1/2}+1$ is recorded on that result page as the corpus's own reading. [MuPo25] is refereed (Invent. Math. 2025).

Fixing $\alpha\le c$ small, the ranges overlap once $p$ is large enough that $\exp(c'(\log p)^{1/3})\ge C_\alpha$, so every $t$ is covered for $p\ge p_0$; none of the four papers makes $p_0$ explicit ("large prime", "$C_\alpha$", "absolute constant $c$", "sufficiently large $N$"). Methods, one sentence each: rectification and induction ([Kr24]); dissociated-set structure and random orderings ([BeKr24], [CoDe26]); anticoncentration of random subset sums ([PhSa26]); absorption and a regularity decomposition of Cayley graphs ([BBKMM25]); the random Hall--Paige theorem through sorting networks ([MuPo25]). Acceptance evidence: [BeKr24], [MuPo25], [CoPe20] and [HOS19] are refereed; [PhSa26], [BBKMM25] and [CoDe26] are preprints, [PhSa26] cited by four 2026 preprints and [BBKMM25] by none in the citation index consulted; the site records the chain in its commentary (5 March 2026) under a label that leaves the problem open, which is not acceptance. Under the preprint qualification the completion for large $p$ is source-supported but not refereed in two of its four links.

The finite remainder. For $p<p_0$ the statement is known only for $t\le12$, $t\ge p-2$ and the $(p-3)$-subsets with nonzero sum, and $p_0$ is not stated, so the finite check the label refers to has no known extent; no source cited here closes any part of it and no proof claim addresses it. This is the sense in which the label DECIDABLE describes the shape of what remains: a statement proved for all sufficiently large primes with the small primes unchecked, not a theorem about every prime.

Adjacent results (context, not the problem). Alspach's conjecture for all finite abelian groups (verified for sets of size up to 11 by Alspach and Liversidge, per [BBKMM25], p. 2); the finite-field model, Theorem 1.3 of [BBKMM25] (every $S\subseteq\mathbb F_2^n\setminus\{0\}$ of size at least an absolute constant has a valid ordering); the approximate version of [BFMPY25]; composite cyclic groups $\mathbb Z_{tp}$ and products of large primes in [CDFL26] (August 2026, abstract only); a 2026 preprint on small sets in abelian groups (arXiv:2603.20961, title only). None concerns the small primes of this problem.

Search scope. None of the routes below found an explicit threshold $p_0$, a treatment of the remaining small primes, a refereed version of [PhSa26], [BBKMM25] or [CoDe26], or a dispute of the chain.

• The site: problem page, discussion thread and proof-claim tab; the formal-conjectures directory listing at its commit of 2026-09-18 (no file); the community database as of 2026-09-18.
• The primary sources at the pages stated: [Kr24] pp. 1--3; [BeKr24] pp. 1--2; [CoDe26] pp. 1--2; [PhSa26] pp. 1--2; [BBKMM25] pp. 1--3, 24--25, 42--43; [MuPo25] pp. 3, 50--52, 57; [CoPe20] pp. 1--2, 6--7; [HOS19] pp. 2, 6, 12--13, 15; [Er73] pp. 126--127; [ErGr80] p. 95; [Gr71] pp. 34--36; [BoHa05] pp. 3--5 and 9--10, with pp. 5--9 for structure.
• arXiv API: the records of the eight arXiv preprints (versions, dates, journal references: only [Kr24]'s and [MuPo25]'s comments and none of the others carry one); the search abs:Graham AND abs:rearrangement AND (abs:"partial sums" OR abs:"valid ordering") sorted by date (two records: [PhSa26] and [CDFL26]).
• Crossref: bibliographic queries for the eight titles (journal records found for [BeKr24], [MuPo25], [CoPe20], [HOS19]; none for [Kr24], [CoDe26], [PhSa26], [BBKMM25]).
• Semantic Scholar citation lists of [PhSa26] (4 records), [BeKr24] (9), [CoDe26] (1) and [BBKMM25] (0), scanned by title.

Not searched: MathSciNet, zbMATH, Google Scholar, X, the MathOverflow post the site and [Kr24] mention. Not held: [ADMS16], [BFMPY25], [CDFL26], the journal texts of [BeKr24], [MuPo25], [CoPe20] and [HOS19].

Remaining gaps. (1) The threshold $p_0$ is not explicit, so the finite remainder has no stated bound; reopening condition: an explicit $p_0$ with a check of the primes below it, or a proof for every prime. (2) Two links of the chain, [PhSa26] and [BBKMM25], and the current small-range record [CoDe26], are preprints; a refereed version of each is the reopening condition for that qualification. (3) The zero-sum $(p-3)$-subsets $\mathbb Z_p\setminus\{0,x,-x\}$, inside the site's range $p-3\le t\le p-1$, are covered by no cited source for a fixed prime: Alspach's conjecture leaves them out, and the appending step of [ADMS16] would need Alspach's conjecture at size $p-4$. [BoHa05]'s Theorem 2 is cited from the paper itself, so the size $p-2$ of Alspach's conjecture is first-hand; the appending step that carries it to $\mathbb Z_p\setminus\{0\}$ is quoted from [CoPe20], p. 7, as [ADMS16] is not held. (4) [Gr71], the source of the problem, is quoted at its Question 10 under Formulation; the paper states the conjecture without proof and without the case $t=p-1$, so Graham's proof of that case, reported by [Er73], is stated in none of the sources cited here. The odd-$n$ directed cycle in the proof of [BoHa05]'s Theorem 2 (p. 4) is an ordering of all of $\mathbb Z_p\setminus\{0\}$ with distinct partial sums, the last being $0$, which is that case; this is the corpus's own reading, recorded on the result page, and not a statement of the paper. (5) Proof coverage is statements only: the result pages record claims checked at statement level, and no argument of the chain has been reviewed; the anticoncentration theorem of [PhSa26] and the absorption argument of [BBKMM25] are the natural candidates for an independent review. (6) The label-versus-statement question above is recorded, not decided.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• graham_1971_sums_integers_taken_fixed_sequence
• graham_1971_sums_integers_taken_fixed_sequence / question_10
• bedert_2024_graham_s_rearrangement_conjecture_beyond_rectification
• bedert_2024_graham_s_rearrangement_conjecture_beyond_rectification / theorem_1_2
• bedert_2025_graham_s_rearrangement_conjecture_over
• bedert_2025_graham_s_rearrangement_conjecture_over / theorem_1_4
• bedert_2025_graham_s_rearrangement_conjecture_over / theorem_7_1
• bode_harborth_2005_directed_paths_diagonals_within_polygons
• bode_harborth_2005_directed_paths_diagonals_within_polygons / theorem_1
• bode_harborth_2005_directed_paths_diagonals_within_polygons / theorem_2
• costa_2020_new_results_about_conjecture_brian_alspach
• costa_2020_new_results_about_conjecture_brian_alspach / corollary_2_5
• costa_2020_new_results_about_conjecture_brian_alspach / corollary_3_2
• costa_2020_new_results_about_conjecture_brian_alspach / corollary_4_3
• costa_2020_new_results_about_conjecture_brian_alspach / corollary_4_4
• costa_2020_new_results_about_conjecture_brian_alspach / proposition_4_2
• costa_2020_new_results_about_conjecture_brian_alspach / theorem_2_4
• costa_2020_new_results_about_conjecture_brian_alspach / theorem_3_1
• costa_2026_new_bounds_weak_sequenceability
• costa_2026_new_bounds_weak_sequenceability / theorem_1_3
• erdos_1973_problems_results_combinatorial_number_theory
• hicks_2019_distinct_partial_sums_cyclic_groups_polynomial
• hicks_2019_distinct_partial_sums_cyclic_groups_polynomial / lemma_4_1
• hicks_2019_distinct_partial_sums_cyclic_groups_polynomial / lemma_4_4
• hicks_2019_distinct_partial_sums_cyclic_groups_polynomial / theorem_2_2
• hicks_2019_distinct_partial_sums_cyclic_groups_polynomial / theorem_2_3
• hicks_2019_distinct_partial_sums_cyclic_groups_polynomial / theorem_3_2
• hicks_2019_distinct_partial_sums_cyclic_groups_polynomial / theorem_4_3
• hicks_2019_distinct_partial_sums_cyclic_groups_polynomial / theorem_4_6
• kravitz_2024_rearranging_small_sets_distinct_partial_sums
• kravitz_2024_rearranging_small_sets_distinct_partial_sums / theorem_1_2
• kravitz_2024_rearranging_small_sets_distinct_partial_sums / theorem_1_3
• muyesser_2022_random_hall_paige_conjecture
• muyesser_2022_random_hall_paige_conjecture / theorem_6_9
• pham_2026_graham_s_rearrangement_conjecture
• pham_2026_graham_s_rearrangement_conjecture / corollary_1_4
• pham_2026_graham_s_rearrangement_conjecture / theorem_1_2
• pham_2026_graham_s_rearrangement_conjecture / theorem_1_3
• erdos_1980_old_new_problems_results_combinatorial_number_theory