{
  "abstract": "The Erdős Matching Conjecture is governed by two competing ways of excluding $s+1$ disjoint edges: one may concentrate all edges on fewer than $k(s+1)$ vertices, or force every edge to meet a fixed $s$-set. We determine a near-optimal range in which the second construction is extremal. For every fixed $k\\ge2$, there is $s_0(k)$ such that, whenever $s\\ge s_0(k)$ and $n\\ge(k+1)s$, every $\\mathcal{F}\\subseteq\\binom{[n]}k$ with $ν(\\mathcal{F})\\le s$ satisfies \\[ |\\mathcal{F}|\\le\\binom nk-\\binom{n-s}k, \\] with equality only for the family of all $k$-sets meeting a fixed $s$-set. This improves the best previous general linear coefficient from $(5k-2)/3$ to $k+1$. In particular, the parameterized form of our argument further lowers the coefficient to $k+0.6$ for $k\\ge5$. Since the two conjectured constructions exchange asymptotic dominance at $n=(ρ_k+o(1))s$ for a coefficient $ρ_k\\in(k,k+1)$, our range lies less than one unit above the unavoidable barrier. We also prove a stability theorem showing that cover families are the only near-extremal configurations throughout this range. A key ingredient in our proof is a probabilistic rigidity statement which forces near-extremal fractional covers to be almost integral.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.19118",
  "author_total": 3,
  "authors": [
    {
      "name": "Mengyu Cao"
    },
    {
      "name": "Hong Liu"
    },
    {
      "name": "Haixiang Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7h08dx4d5v24f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.19118",
  "fulltext_url": null,
  "id": "7h08dx4d5v24f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.19118",
  "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": "W7203930432",
  "pdf_url": "https://arxiv.org/pdf/2608.19118",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7h08dx4d5v24f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2608.19118",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.19118"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A Near-Optimal Linear Range for the Erdős Matching Conjecture",
  "title_text": "A Near-Optimal Linear Range for the Erdős Matching Conjecture",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/7h08dx4d5v24f",
  "venue": "arXiv",
  "versions": []
}