FormalConjectures.ErdosProblems.«348»

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import FormalConjecturesUtil

Erdős Problem 348

Reference: erdosproblems.com/348


namespace Erdos348


open Erdos348

For what values of $0 \leq m < n$ is there a complete sequence $A = {a_1 \leq a_2 \leq \cdots}$ of integers such that

$A$ remains complete after removing any $m$ elements, but

$A$ is not complete after removing any $n$ elements.

@[category research open, AMS 11]
theorem declaration uses 'sorry'erdos_348 :
    { (m, n) | (m) (n) (_ : m < n) (a : ℕ → ℕ) (_ : Monotone a)
      (_ : ∀ s, s.card = m → IsAddComplete (Set.range (Function.updateFinset a s 0)))
        (_ : ∀ t, t.card = n → ¬IsAddComplete (Set.range (Function.updateFinset a t 0))) } =
    answer(sorry) := by⊢ {x |
    ∃ m n,
      ∃ (_ : m < n),
        ∃ a,
          ∃ (_ : Monotone a) (_ :
            ∀ (s : Finset ℕ), s.card = m → IsAddComplete (Set.range (Function.updateFinset a s 0))) (_ :
            ∀ (t : Finset ℕ), t.card = n → ¬IsAddComplete (Set.range (Function.updateFinset a t 0))), (m, n) = x} =
  sorry
  sorryAll goals completed! 🐙


end Erdos348