Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 274

claims/: The 6 claim pages of Problem 274, one per claimant's result; the problem's standing derives from them.

Statement. If $G$ is a group then can there exist an exact covering of $G$ by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets)

Formulation. An exact covering here is a partition of $G$ into finitely many left cosets, as in the Herzog–Schönheim conjecture that the site's commentary states. "Different sizes" means pairwise different sizes. Read as "not all the same size", the question is trivially yes: in $\mathbb Z/4$ , the coset $\{0,2\}$ and the singletons $\{1\}$ and $\{3\}$ form such a partition. Erdős asked the question for abelian groups in [Er77c] and [ErGr80], as the site's commentary records. He asked it for finite groups that need not be abelian in [Er97c] (P. Erdős, Some of my favorite problems and results, The Mathematics of Paul Erdős I (1997), 47–67, p. 53), as the site's commentary said before an edit in late October 2025 (history). That edit followed a forum comment of 2025-10-07 noting that the abelian case is settled, and it dropped the word abelian. The site's statement now concerns every group, and the page's standing concerns that statement. In a partition of a group into finitely many cosets, every subgroup has finite index (Korec and Znám, Theorem 1). In an infinite group all the cosets therefore have the cardinality of the group, so read with sizes, the answer there is no. For a finite group, cosets of different sizes are cosets of subgroups of different indices. The statement is therefore equivalent to the Herzog–Schönheim conjecture for finite groups. Read with indices, that case gives the conjecture for all groups, by passing to the finite quotient by the common core of the subgroups. Erdős's abelian question has the answer no: see Berger, Felzenbaum and Fraenkel for finite nilpotent groups and Sun for subnormal subgroups.

Status. Open.

Source. erdosproblems.com/274, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #274, https://www.erdosproblems.com/274.

References.

[Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[MaSc19] Margolis, Leo and Schnabel, Ofir, The Herzog-Schönheim conjecture for small groups and harmonic subgroups. Beitr. Algebra Geom. (2019), 399-418.
[Su04] Sun, Zhi-Wei, On the Herzog-Schönheim conjecture for uniform covers of groups. J. Algebra (2004), 153-175.

Formalization. Statement in formal-conjectures.

Current assessment

The arbitrary-group Herzog--Schönheim conjecture remains open. The abelian case, which Erdős asked in [Er77c] and [ErGr80], follows from Sun's Theorem 1.1 [Su04] for cosets of subnormal subgroups (claim page), and Margolis and Schnabel's Theorem A [MaSc19] verifies the conjecture for groups of order below $1440$ (claim page); both statements are recorded on the library cards, and neither proof has been reviewed here. For finite abelian groups the case was settled earlier by Berger, Felzenbaum and Fraenkel for all finite nilpotent groups.

Akman--Sissokho proved that no counterexample can use at most seven cells (claim page). An unreviewed computer-assisted proof candidate by Itabe claims the stronger bound of seventeen cells: every counterexample would need at least eighteen. It is the site's one proof claim, submitted 2026-08-17, and is recorded as claimed on its claim page. Its exact release, includes a complete Lean 4 formalization, but neither the mathematical proof nor the formalization has been independently verified in this repository, and no Lean build has been run here.

Three further results settle other classes of finite groups and have no claim pages: Berger, Felzenbaum and Fraenkel's theorem for pyramidal groups (Fund. Math. 128 (1987)), which include the supersolvable groups; Ginosar and Schnabel's theorems for groups whose orders have at most two prime divisors, or three not including both $2$ and $3$ , and for $A_5$ , $S_5$ , $\mathrm{Sz}(8)$ and $\mathrm{Sz}(32)$ (J. Comb. Number Theory 3 (2011)); and Garonzi and Margolis's preprint for simple and symmetric groups (arXiv:2509.25118). They are cited through their library cards.

As of 2026-10-05, the site shows OPEN (last edited 31 October 2025) with Itabe's partial claim as its one proof claim; the community database says open; formal-conjectures tags erdos_274 and herzog_schonheim research open, only the abelian variant carrying a formal-proof link (pull request 4415, merged 2026-07-21), a formalization of the abelian case of Berger, Felzenbaum and Fraenkel's theorem; conjectures.io keeps a live bounty whose one partial contribution of 19 September 2026 says the general conjecture remains open, and it has no claim page because it asserts no new result. A machine-checked, unrefereed partial result is the Palomar registry entry PALOMAR-2026-10-02-000004 (registered 2 October 2026), Murali Menon's Lean 4 proof of the catalog's statement with the hypothesis that $G$ is solvable added, finite or infinite $G$ , developed with Claude and OpenAI Codex models; it is recorded as claimed on its claim page. Its only review is automated, no human expert has reviewed it, its informal write-up is not public, and the only earlier solvable-case claim (arXiv:1901.10131) was withdrawn in 2019. This is a different work from the coset-partition note on $A_5$ . It was neither built nor examined here. If the registered proof stands, solvable groups leave the remaining problem stated below; the status is unchanged.

The remaining problem is to prove the repeated-index conclusion without a cell bound or to construct and verify a counterexample with at least eighteen cells.

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.

Linked from (75)

Berger, Felzenbaum and Fraenkel's finite nilpotent groups Sun's theorem for subnormal subgroups Margolis and Schnabel's groups of order below 1440 Akman and Sissokho's bound of eight cells Itabe's bound of eighteen cells Menon's Lean proof for solvable groups problems/covering_systems/E0274/claims Problem 947 The Mirsky–Newman theorem that no exact covering system has distinct moduli Covering Systems Covering Systems covering_systems/margolis_2019_herzog_schonheim_conjecture_small_groups Lemma 2.3: arithmetic conditions on a coset partition without multiplicity Proposition 4.2: (2r1, 2r2, 2r3) with pairwise coprime ri is never G-harmonic Proposition 4.3: (3r1, 3r2, 3r3, 3r4) with pairwise coprime ri is never G-harmonic Proposition 4.5: (2r1, 4r2, 4r3, 4r4) with pairwise coprime ri and r1 odd is never G-harmonic Proposition 4.7: (3, 3r2, 6r3, 6r4, 6r5) with pairwise coprime ri and r2 odd is never G-harmonic Theorem A: the Herzog-Schönheim conjecture for groups of order below 1440 Theorem B: G-harmonic tuples of length at most four are Z-harmonic covering_systems/sun_2004_herzog_schonheim_conjecture_uniform_covers Corollary 4.2: two equal indices divisible by a large prime Theorem 1.1: a nontrivial uniform coset cover by subnormal subgroups repeats an index Theorem 4.1: a p-adic multiplicity bound for uniform coset covers Theorem 4.3: index multiplicities and prime divisors in uniform coset covers Steiner Coset Partitions of Groups Transversal coset partitions of groups Corollary 10 (p. 4): finite and infinite groups realize the same index lists Theorem 2 (p. 2): transversal coset partitions for two subgroups Theorem 3 (p. 2): transversal coset partitions for three commuting subgroups Theorem 4 (p. 3): no pure partition by three subgroups unless all have index 3 Theorem 5 (p. 3): four commuting subgroups, one of index 2 Theorem 6 (p. 3): Herzog–Schönheim for at most seven distinct subgroups Theorem 7 (p. 3): nonstandard transversal partitions of direct products Steiner coset partitions for five mutually commuting subgroups Proposition 3 (p. 2): five DPMC subgroups with G ≠ HKLMN Theorem 4 (p. 3): no Steiner partition when G = HKLMN Steiner Coset Partitions of Groups – ERRATUM Berger et al.: The Herzog-Schönheim Conjecture for Finite Nilpotent Groups Corollary IV (p. 333): a partition of a finite nilpotent group into at least two cosets has two cosets of the same order Remark (p. 333): a partition as in Theorem III has N+1 parts of equal size Theorem 1 (p. 329): a union of product sets is at least as large as the union of their equivalent boxes Theorem II (p. 331): the size of a union of prime-power boxes is a sum of Euler's function over divisors Theorem III (p. 331): a partition of a prime-power box into prime-power product sets has two parts of equal size Berger et al.: Remark on the multiplicity of a partition of a group into cosets The Herzog-Schönheim conjecture for simple and symmetric groups Lemma 2.1 (p. 3): a finite group with reciprocal index sum below 2 satisfies Herzog-Schönheim Proposition 3.2 (p. 7): reciprocal index sums of symmetric and alternating groups Remark 5.1 (p. 19): the reciprocal-sum method fails for almost simple groups and products Theorem 1.2 (p. 2): the Herzog-Schönheim conjecture holds for symmetric groups Theorem 1.3 (p. 2): the Herzog-Schönheim conjecture holds for simple groups

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