erdős #274 · Herzog-Schönheim conjecture
If is a group then can there exist an exact covering of by more than one cosets of different sizes? (i.e. each element is contained in exactly one of the cosets)
Worked, still open.
group theory · open · formalized (Lean) · 0 attempts
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unverified AI candidates (2)
gpt-erdos · GPT-5.2 Pro + Deep Research · unverified
A coset $gH$ always has the same size as the subgroup $H$. So “different sizes” just means you are using cosets of different subgroups (with different orders).
candidate solution ↗llm-hunter · gpt 5.2, gpt pro 5.2 · unverified
2 LLM attack(s) recorded (gpt 5.2, gpt pro 5.2); unverified.
candidate solution ↗formal
AMS 20 · open (literature)
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)formal-conjectures/274.lean ↗links
question of Herzog and Schönheim · reference
subnormal · reference
status
open