{
  "abstract": "We prove that the pairs of integers $0\\le m<n$ for which a nondecreasing integer sequence can remain complete after every deletion of $m$ terms and become incomplete after every deletion of $n$ terms are exactly those with $m\\le1$. Here a sequence is complete if every sufficiently large integer is a finite sum of terms with distinct indices. This answers Erdős Problem 348, posed by Graham and later included in the book of Erdős and Graham. The proof uses a central-interval theorem: if a complete nondecreasing positive integer sequence $(a_i)$ has prefix sums $S_j$ with $S_j-a_{j+1}\\to\\infty$, then each sufficiently long prefix represents every integer from any fixed completeness threshold $T$ to $S_j-T$. We also refute Graham's conjecture, later repeated by Erdős and Graham, that $(\\lfloor tγ^n\\rfloor)_{n\\ge1}$ is complete for every $t>0$ and $1<γ<(1+\\sqrt5)/2$. We obtain the counterexample by combining Dubickas's fractional-part theorem with an elementary sign adjustment. For a common base in this range, we further construct two such sequences whose interleaving is incomplete and whose coefficient ratio is not a rational multiple of any integer power of the base.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25107",
  "author_total": 1,
  "authors": [
    {
      "name": "Jesse Geneson"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/00t9zxq54zm4g.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25107",
  "fulltext_url": null,
  "id": "00t9zxq54zm4g",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25107",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2602|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7214123129",
  "pdf_url": "https://arxiv.org/pdf/2609.25107",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/00t9zxq54zm4g.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25107",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25107"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Deletion thresholds and exponential examples for complete sequences",
  "title_text": "Deletion thresholds and exponential examples for complete sequences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/00t9zxq54zm4g",
  "venue": "arXiv",
  "versions": []
}