Bedert 2025 graham s rearrangement conjecture over

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theorem_1_4: The large range of the valid-ordering problem in every finite group: subsets of size at least |G|^{1-c} for an absolute c > 0 have orderings with distinct partial products, by absorption and a Cayley-graph regularity decomposition.

theorem_7_1: The very large range of the valid-ordering problem, attributed to Müyesser and Pokrovskiy and proved in the paper's Appendix A from their random Hall–Paige machinery: subsets missing at most N^{1-gamma} elements have orderings with distinct partial products.

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Benjamin Bedert, Matija Bucić, Noah Kravitz, Richard Montgomery, Alp Müyesser, On Graham's rearrangement conjecture over $\mathbb{F}_2^n$. arXiv:2508.18254 (2025).

Theorem 1.3 shows there is an absolute constant C such that every S ⊆ F_2^n
{0} with |S| >= C admits a valid ordering (all partial sums distinct), an essentially complete resolution of the valid-ordering question for the groups F_2^n and hence of the finite-field-model version of Graham's conjecture; the authors note the same method handles finite abelian groups of bounded exponent, e.g. F_p^n for fixed p. Theorem 1.4 answers the question in every finite group once the subset is moderately large: there is c > 0 such that in every finite, possibly nonabelian, group G every S ⊆ G \ {id} with |S| >= |G|^{1-c} has a valid ordering, a large improvement on Müyesser–Pokrovskiy's (1-o(1))|G| threshold. The proof splits into sparse and dense regimes, combining additive and probabilistic combinatorics: Freiman–Ruzsa for the sparse F_2^n case (exploiting the abundance of small zero-sum subsets in moderately dense subsets of F_2^n) and the absorption method plus a regularity-type structural result decomposing any Cayley graph into mildly quasirandom components for the dense case. It bears on problem 475 (Graham's rearrangement conjecture over F_p) by resolving the analogous question in the finite-field model and by pushing the dense-set case of the original conjecture from (1-o(1))p down to p^{1-c}; the paper leaves the intermediate range of sizes over F_p open.

The retained folder-name PDF is arXiv:2508.18254v1 (25 August 2025, 43 pp.; "40 pages" in the arXiv comment), whose pagination is used here; no journal record was found (Crossref bibliographic query, 2026-09-18): a preprint. Read status: claims checked for Question 1.1, Conjecture 1.2, Theorems 1.3, 1.4, 1.5 and 7.1, Lemma A.1 and Theorem A.2 (pp. 2--3, 25, 42, text layer) on 2026-09-18; the proof of Theorem A.2 (pp. 42--43) was read for structure and the other proofs were not read. Theorem 7.1, the extremely dense case $|S|\ge N-N^{1-\gamma}$, is labeled "([36])" and proved in Appendix A from Lemma 6.22 and the method of Theorem 6.9 of Müyesser and Pokrovskiy; it is the explicit form of the site's very large range for Problem 475. Result pages: theorem_1_4 and theorem_7_1.

Source: https://arxiv.org/abs/2508.18254. The arXiv record (https://arxiv.org/abs/2508.18254, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #475

Results to transcribe.

• Theorem 1.3: An absolute constant C exists such that every S ⊆ F_2^n \ {0} with |S| >= C has a valid ordering; the method extends to abelian groups of bounded exponent.
• Theorem 1.4: There is c > 0 such that for every finite (possibly nonabelian) group G, every S ⊆ G \ {id} of size at least |G|^{1-c} admits a valid ordering.
• Structural ingredient: A regularity-lemma-style decomposition of any Cayley graph into mildly quasirandom components, used to drive the dense-case absorption argument.