Erdős Problem #274

If G is a group, can there exist an exact covering of G by more than one coset of different sizes? (i.e. each element is contained in exactly one of the cosets.) The conjectured answer is no: in every such exact covering, two of the subgroups have the same cardinality.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-erdos-274,
  title        = {Erdős Problem #274},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-274}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

erdos_274. If $G$ is a group, can there exist an exact covering of $G$ by more than one coset of different sizes? (i.e. each element is contained in exactly one of the cosets.)

The conjectured answer is no: in every such exact covering, two of the subgroups have the same cardinality.

herzog_schonheim. Let $G$ be a group, and let $A = {a_{1} G_{1}, \dots, a_{k} G_{k}}$ be a finite system of left cosets of subgroups $G_{1}, \dots, G_{k}$ of $G$ .

Herzog and Schönheim conjectured that if $A$ forms a partition of $G$ with $k > 1$ , then the indices $[G : G_{1}], \dots, [G : G_{k}]$ cannot be distinct.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«274» (2 statements). answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_274 : answer(sorry) ↔ ∃ (G : Type*) (_ : Group G),
    1 < ENat.card G ∧ ∃ (ι : Type*) (_ : Fintype ι) (P : Group.ExactCovering G ι),
    1 < Fintype.card ι ∧ ∀ i j, i ≠ j → #(P.parts i) ≠ #(P.parts j)

theorem herzog_schonheim {G : Type*} [Group G] (hG : 1 < ENat.card G) {ι : Type*} [Fintype ι]
    (hι : 1 < Fintype.card ι) (P : Group.ExactCovering G ι) :
    ∃ i j, i ≠ j ∧ (P.parts i).index = (P.parts j).index

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/274. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.

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