{
  "abstract": "We prove the Erdős Matching Conjecture for $4$-uniform hypergraphs for every matching number $s\\ge6004$. Specifically, for integers $s\\ge 6004$ and $n\\ge 4s+4$, if a $n$-vertex 4-uniform hypergraph $\\mathcal F$ does not contain a matching of size $s+1$, then \\[ |\\mathcal F|\\le \\max\\left\\{\\binom{4s+3}{4},\\binom n4-\\binom{n-s}{4}\\right\\}. \\] Our proof introduces a finite-board reduction for general uniformity. It reduces the global conjecture to a lower-uniformity bound and a fixed finite optimization at the two adjacent vertex numbers where the candidate constructions exchange dominance. In the $4$-uniform case the board has $19$ vertices, and its weighted inequality splits into $19$-, $15$-, and $11$-vertex layers. The hardest layer is resolved by exact rational dual certificates and deterministic integer searches. All computer-assisted steps are checked in exact arithmetic by verifiers that reconstruct the finite systems directly from their mathematical definitions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.26060",
  "author_total": 3,
  "authors": [
    {
      "name": "Jianfeng Hou"
    },
    {
      "name": "Caiyun Hu"
    },
    {
      "name": "Xizhi Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e1ng816g4p8vb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.26060",
  "fulltext_url": null,
  "id": "e1ng816g4p8vb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.26060",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7162407586",
  "pdf_url": "https://arxiv.org/pdf/2605.26060",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e1ng816g4p8vb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2605.26060",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.26060"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Erdős Matching Conjecture for 4-uniform hypergraphs",
  "title_text": "The Erdős Matching Conjecture for 4-uniform hypergraphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/e1ng816g4p8vb",
  "venue": "arXiv",
  "versions": []
}