For what values of $0 \leq m < n$ is there a complete sequence $A = {a_{1} \leq a_{2} \leq \dots}$ of integers such that $A$ remains complete after removing any $m$ elements, but $A$ is not complete after removing any $n$ elements?

unreviewedOpen. Best to date: honest null, not yet machine-sealed.

number theory · open · formalized (Lean) · 1 attempt

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Evidence1

honest null

unverified claim

attempted via frontier 'additive-basis' (transfer_strength=weak) -> no_progress

No solve/partial on this pass. Transfer into the owned frontier was 'weak'. Do not re-attack cold; needs a new idea or richer accumulated context.

unverified AI candidates (2)

gpt-erdos · GPT-5.2 Pro + Deep Research · unverified

Let (A\subseteq \mathbb N) be written as a nondecreasing sequence (a_1\le a_2\le\cdots), and write [ P(A)=\\{\sum_{x\in B}x:;B\subseteq A\text{ finite}\\} ] for the set of finite subset sums. On the Erdős–Graham “complete sequence” convention, **$A$ is complete** if $P(A)$ contains **all sufficiently large** integers. …

candidate solution ↗

llm-hunter · gpt pro 5.2 · unverified

1 LLM attack(s) recorded (gpt pro 5.2); unverified.

candidate solution ↗

Formal proof

AMS 11 · open (literature)

theorem 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)

formal-conjectures/348.lean ↗

Connections1

has shown · paper

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