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Erdős problem 274 - herzog schonheim Let G G G be a group, and let A = { a 1 G 1 , … , a k G k } A = \{a_1G_1, \dots, a_kG_k\} A = { a 1 G 1 , … , a k G k } be a finite system of left cosets of
subgroups
G 1 , … , G k G_1, \dots, G_k G 1 , … , G k of
.
Herzog and Schönheim conjectured that if
forms a partition of
with
, then the
indices
[ G : G 1 ] , … , [ G : G k ] [G:G_1], \dots, [G:G_k] [ G : G 1 ] , … , [ G : G k ] cannot be distinct.
Two ways to claim this Each is a separate task with its own bundle and its own bounty. Pick the one your proof argues for.
Bounty
$4,761
paid on an accepted proof
Set by bounty policy dynamic-age-v1: the amount is worked out from how long the problem has stood open, so it moves as the pool and the pool's age profile move.
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Lean type
∀ {G : Type u_1} [inst : Group G],
1 < ENat.card G →
∀ {ι : Type u_2} [inst_1 : Fintype ι],
1 < Fintype.card ι →
∀ (P : Erdos274.Group.ExactCovering G ι), ∃ i j, i ≠ j ∧ (P.parts i).index = (P.parts j).indexWhat you must prove
import FormalConjectures.ErdosProblems.«274»
import TaskSupport
namespace Bounty
theorem target : ¬ (fcTypeOfName% "Erdos274.herzog_schonheim") := by
sorry
end Bounty
Pinned source: FormalConjectures/ErdosProblems/274.lean
Source type SHA-256 sha256:5bb3763bc9a5ca0a8e1c49e6e598ca69f347187c0bdb626db8a8c9786d54d727
Task id fc-379fc029-erdos274-herzog-schonheim-55915c24e0-counterexample-v1
Task commitment sha256:fcfc34d36275e9efe049edacf376c2fbc8bc05410c90d4a8c404a2f701bbc225 Something wrong with this formalization?
A statement that does not faithfully capture the original conjecture is the one real risk here, so we would rather hear about it early - before someone spends weeks on it.
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Erdős problem 274 - herzog schonheim · Conjectures.io