FormalConjectures.ErdosProblems.«653»

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import FormalConjecturesUtil

Erdős Problem 653

Reference: erdosproblems.com/653


open Finset EuclideanGeometry Filter


namespace Erdos653

Let $x_1,\ldots,x_n\in \mathbb{R}^2$ and let $R(x_i)=#{ \lvert x_j-x_i\rvert : j\neq i}$, where the points are ordered such that $$R(x_1)\leq \cdots \leq R(x_n).$$ Let $g(n)$ be the maximum number of distinct values the $R(x_i)$ can take. Is it true that $g(n) \geq (1-o(1))n$?

@[category research open, AMS 5 52]
theorem declaration uses 'sorry'erdos_653 : answer(sorry) ↔ ∃ o : ℕ → ℝ, o =o[atTop] (1 : ℕ → ℝ) ∧
    ∀ᶠ n in atTop, (1 - o n) * n ≤ maximalDistinctDistancesFrom n := by⊢ True ↔ ∃ o, o =o[atTop] 1 ∧ ∀ᶠ (n : ℕ) in atTop, (1 - o n) * ↑n ≤ ↑(maximalDistinctDistancesFrom n)
  sorryAll goals completed! 🐙


end Erdos653