Problem 348

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claims/: The 1 claim page of Problem 348, one per claimant's result; the problem's standing derives from them.

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Statement. 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?

Formulation. The site defines a set as complete when its finite subset sums contain all sufficiently large integers (its definitions page), and the statement lists $A$ as $a_1\le a_2\le\cdots$, so equal values are separate terms, as the remarks' Fibonacci example $1,1,2,\ldots$ needs. The remarks contrast the strong sense, in which every positive integer must be a subset sum and van Doorn excluded every $m\ge2$, and say that Erdős and Graham most likely meant the eventual sense. The page reads the problem in the eventual sense with repeated values, the reading of Geneson's claim; van Doorn's strong-sense result settles no instance of it and has no claim page.

Status. Claimed, against the site's label OPEN. One pending full claim is recorded: [[problems/additive_bases/E0348/claims/2026_09_09_geneson|Geneson's classification of deletion thresholds]], a 2026 preprint stating that the pairs asked for are exactly those with $m\in\{0,1\}$; the frontmatter standing, claimed/answered, follows from it, and the site's remarks record the case $m=2$, $n=3$ as unknown.

Source. erdosproblems.com/348, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #348, https://www.erdosproblems.com/348.

Formalization. Statement in formal-conjectures.

Current assessment

The site records Problem 348 as OPEN,. The frontmatter standing, claimed/answered, is derived from the one claim page below: Geneson's 2026 preprint asserts the full classification, exactly $m\le1$, and no acceptance of it is recorded. This page records no literature search beyond the site's thread and the preprint's card and no independent assessment of proof coverage.

Progress

Geneson's Theorem 1 is recorded on its library card, linked below.

Known Results

One pending full claim. Jesse Geneson's preprint (arXiv:2609.25107, 2026-09-20; its earlier ResearchGate note was submitted to the site's proof-claims thread on 2026-09-09) claims that the pairs asked for are exactly those with $m\in\{0,1\}$, reading completeness in the eventual sense with repeated values allowed: every two-term deletion preserving completeness forces some deletion of every finite size to preserve it, so no $m\ge2$ works, while the powers of $2$ and the Fibonacci sequence give $m=0$ and $m=1$. The claim page Geneson's classification records the statement, the author's disclosure of machine assistance, the site's OPEN label, and a third-party Lean proof of the deletion lemma only; the preprint's card and its Theorem 1 page are the library links below. The reading adopted, and van Doorn's strong-sense exclusion of $m\ge2$ that the site's remarks record, are stated in the Formulation above.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• geneson_2026_deletion_thresholds_exponential_examples_complete_sequences
• geneson_2026_deletion_thresholds_exponential_examples_complete_sequences / theorem_1