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Claim. Call a sequence $A=(a_i)_{i\ge1}$ of integers complete when every sufficiently large integer is a sum of terms with distinct indices, each occurrence used at most once, and let a deletion remove occurrences. For integers $0\le m<n$ there is a nondecreasing integer sequence that stays complete after every deletion of $m$ occurrences and is incomplete after every deletion of $n$ occurrences if and only if $m\le1$. So the pairs Problem 348 asks for are exactly those with $m\in\{0,1\}$: the powers of $2$ give $m=0$, $n=1$, and the Fibonacci sequence $1,1,2,3,5,\ldots$ gives $m=1$, $n=2$, while no sequence works for any $m\ge2$. The result is Theorem 1 of Geneson, J., Deletion thresholds and exponential examples for complete sequences, arXiv:2609.25107 (v1 2026-09-20), whose statement the card Theorem 1 records clause by clause. The ResearchGate note of the forum claim, "Deletion thresholds for complete sequences", is the result's earlier posting; its posting date is not recorded, so the page is dated by the forum claim.

Submission note. Posted to erdosproblems.com as a proof claim by Jesse Geneson (account jtg) on 9 September 2026, giving "GPT-6 Astra, Claude Opus 5" as the AI used:

I prove that only $m=0,1$ work. If every two-term deletion preserves completeness, then some deletion of any finite size does too. Notes: Here completeness means that every sufficiently large integer is a sum of finitely many terms, with each occurrence used at most once. The proof was found by Codex with GPT-6 Astra Ultra. I directed the writing and auditing workflow, then read and edited the proof and used GPT-6 and Claude Opus 5 for additional audits.

Formulation. The claim reads "complete" in the eventual sense, finitely many exceptions allowed, and allows repeated values as separate occurrences. The site's remarks record that van Doorn excluded every $m\ge2$ under the strong requirement that every positive integer be a subset sum, and say that Erdős and Graham most likely meant the eventual notion; the claim settles that eventual reading, which is the reading the problem's standing concerns here.

Argument. As the paper and its card describe it: a central-interval theorem (Theorem 2) shows that a complete nondecreasing positive sequence whose prefix slack $S_j-a_{j+1}$ tends to infinity represents, by its first $N$ terms, every integer between the completeness threshold $T$ and $S_N-T$ for every large $N$; a sequence that stays complete after every single deletion has slack tending to infinity (Lemma 3). From these, if every two-term deletion preserves completeness then so does some deletion of any prescribed finite size, which excludes $m\ge2$; the two constructions give $m=0$ and $m=1$.

Standing. Claimed. The paper is a preprint with no journal record on the arXiv page the site's label is OPEN and no outside review of the proof is known. The author submitted the claim to the proof-claims thread on 2026-09-09 (the discussion link), stating that the proof was found by Codex with GPT-6 Astra Ultra, that they directed the writing and auditing, read and edited the proof, and used GPT-6 and Claude Opus 5 for further audits; the paper's closing declaration states that the proofs were found with the assistance of Codex with GPT-6 Astra Ultra, that the author directed separate writing and auditing teams, read and edited the proofs and requested further revisions, and that they take responsibility for the content; it does not mention the GPT-6 and Claude Opus 5 audits. In the thread the forum user Woett observed on 2026-09-10 that the argument gives the following: if $A$ is a nonempty set of positive integers with $A\setminus\{a,b\}$ complete for all $a,b\in A$, then some infinite $B\subseteq A$ has $A\setminus B_0$ complete for every finite $B_0\subseteq B$; Woett linked a Lean 4 file (the formalization link, pinned to its last commit of 2026-09-10) proving the weaker statement that for every $k$ some $k$-element $S\subseteq A$ leaves $A\setminus S$ complete. The file imports Mathlib, credits the formalization to the system Aristotle from Harmonic, and prints the axioms of its theorem main; it covers the deletion lemma that excludes $m\ge2$ for sets, not the constructions and not Theorem 1 itself. The claim lists no evidence: the paper is unrefereed, no outside review is known, and the Lean file covers only the lemma.

Depends on. Nothing in this wiki.