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Erdős problem 653 Let x 1 , … , x n ∈ R 2 x_1,\ldots,x_n\in \mathbb{R}^2 x 1 , … , x n ∈ R 2 and let R ( x i ) = # { ∣ x j − x i ∣ : j ≠ i } R(x_i)=\#\{ \lvert x_j-x_i\rvert : j\neq i\} R ( x i ) = # {∣ x j − x i ∣ : j = i } ,
where the points are ordered such that
R ( x 1 ) ≤ ⋯ ≤ R ( x n ) . R(x_1)\leq \cdots \leq R(x_n). R ( x 1 ) ≤ ⋯ ≤ R ( x n ) .
Let
be the maximum number of distinct values the
can take. Is it true that
g ( n ) ≥ ( 1 − o ( 1 ) ) n g(n) \geq (1-o(1))n g ( n ) ≥ ( 1 − o ( 1 )) n ?
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
$770
paid on an accepted proof
Set by bounty policy dynamic-age-v2-locked: 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.
Submit a proof Your file is checked for free before any credit is spent.
Lean type
True ↔
∃ o,
o =o[Filter.atTop] 1 ∧
∀ᶠ (n : ℕ) in Filter.atTop, (1 - o n) * ↑n ≤ ↑(EuclideanGeometry.maximalDistinctDistancesFrom n)What you must prove
import FormalConjectures.ErdosProblems.«653»
import TaskSupport
namespace Bounty
theorem target : ¬ (fcTypeOfName% "Erdos653.erdos_653") := by
sorry
end Bounty
Pinned source: FormalConjectures/ErdosProblems/653.lean
Source type SHA-256 sha256:d4ce1dcee1ba887c8f51c8a710ae1fbb59bb0de3a7113d7ef121c56f19a762d1
Task id fc-8432eac9-erdos653-erdos-653-42c9d5c2c1-counterexample-v1
Task commitment sha256:96abc5f642ce1c4cef95c06d5209aeaae7718d707e54fc4ec6ed9eecb0264695 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 653 · Conjectures.io