348.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 348

References:

erdosproblems.com/348
[Ge26] Geneson, J., Deletion thresholds and exponential examples for complete sequences. arXiv:2609.25107 (2026).

Imports (1)

FormalConjecturesUtil

theorem Erdos348.erdos_348 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 1 sorry line 45 source
Source documentation
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
1. $A$ remains complete after removing any $m$ elements, but
2. $A$ is not complete after removing any $n$ elements.
Geneson [Ge26, Theorem 1] proved that the pairs are exactly those with $m \leq 1$. Completeness allows finitely many exceptions, and deletions remove indexed occurrences.
depends on sorry what this means
Statement uses: Eq, Set, Prod, Nat, Set.ofPred, Exists, LT.lt, Monotone, Finset, Finset.card, IsAddCompleteNatSeq', Function.updateFinset, OfNat.ofNat, Subtype, Membership.mem, Not, Prod.mk, Erdos348.erdos_348.match_1, And, LE.le
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_348 :
    { (m, n) | (m) (n) (_ : m < n) (a : ℕ → ℕ) (_ : Monotone a)
      (_ : ∀ s, s.card = m → IsAddCompleteNatSeq' (Function.updateFinset a s 0))
        (_ : ∀ t, t.card = n → ¬IsAddCompleteNatSeq' (Function.updateFinset a t 0)) } =
    answer({(m, n) : ℕ × ℕ | m < n ∧ m ≤ 1}) := by
…
```
Mentions: IsAddCompleteNatSeq'