Eventual existence of tail three-family partition

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Erdos390.eventual_tail_three_family_partition_exists

by doctosil · Sep 6, 2026 · Mathlib c5ea003 (Lean v4.30.0)

asymptoticscombinatoricsnumber-theory

Fix a constant $c > C_0 = \frac{4029639598}{25970038185}$ and let $h = \left\lceil c \frac{n}{\log n} \right\rceil$ . For all sufficiently large $n \in \mathbb{N}$ , given any divisor $D \in \mathbb{N}$ and central subset $\mathrm{central} \subseteq (n, 2n]$ such that $\prod \mathrm{central} = \binom{2n}{n} D$ and $D \mid \prod_{(2n, 2n + h]} a$ , there exist pairwise disjoint finite subsets $\mathrm{fixed}, \mathrm{bank}, \mathrm{candidates} \subseteq (2n, 2n + h]$ satisfying:

\left(\prod_{a \in \mathrm{fixed}} a\right) \cdot \left(\prod_{b \in \mathrm{bank}} b\right) \cdot \left(\prod_{c \in \mathrm{candidates}} c\right) \cdot D = \prod_{k \in (2n, 2n + h]} k.

Preamble

import Definitions.Def_erdos390_problem
open Filter
open Erdos390

Formal statement

namespace Erdos390

theorem eventual_tail_three_family_partition_exists :
    ∀ c : ℝ, C0 < c →
      ∀ᶠ n : ℕ in atTop,
        ∀ (D : ℕ) (central : Finset ℕ),
          central ⊆ factorInterval n (2 * n) →
          central.prod id = Nat.choose (2 * n) n * D →
          D ∣ (factorInterval (2 * n) (2 * n + Nat.ceil (c * secondOrderScale n))).prod id →
          ∃ (fixed bank candidates : Finset ℕ),
            fixed ⊆ factorInterval (2 * n) (2 * n + Nat.ceil (c * secondOrderScale n)) ∧
            bank ⊆ factorInterval (2 * n) (2 * n + Nat.ceil (c * secondOrderScale n)) ∧
            candidates ⊆ factorInterval (2 * n) (2 * n + Nat.ceil (c * secondOrderScale n)) ∧
            Disjoint fixed bank ∧
            Disjoint fixed candidates ∧
            Disjoint bank candidates ∧
            fixed.prod id * bank.prod id * candidates.prod id * D =
              (factorInterval (2 * n) (2 * n + Nat.ceil (c * secondOrderScale n))).prod id := by sorry

end Erdos390

Source

Shouqiao Wang, A Proposed Solution to Erdős Problem 390, Section 10, BankPaperGuardedUpperProductAssembly.lean

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