Erdős Problem 274

formal-conjectures · erdos-problems · ams-20

Let $G$ be a group, and let $A = \{a_1G_1, \dots, a_kG_k\}$ 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.

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://www.erdosproblems.com/274
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://en.wikipedia.org/wiki/Herzog%E2%80%93Sch%C3%B6nheim_conjecture
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/1803.08301
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/1803.03569
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://pmc.ncbi.nlm.nih.gov/articles/PMC7247885/
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://arxiv.org/abs/1804.11103
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/ErdosProblems/274.lean
Local target: Corpus.Erdos274.herzog_schonheim
Source SHA-256: 2414e1c69eeec6ac40b58c59a63eafa6006e7660b672638db2a60adea24c1974
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record