{
  "abstract": "We resolve a conjecture of Łuczak, Mieczkowska and Šileikis on the maximal tail probability of sums of independent and identically distributed nonnegative random variables. Specifically, for every integer $k\\ge1$ and every $0\\le x\\le1/k$, we prove \\[ \\sup_{\\mathbb E X_1\\le x}\\mathbb{P}(X_1+\\cdots+X_k\\ge1) =\\max\\{1-(1-x)^k,(kx)^k\\}, \\] where the supremum is over nonnegative i.i.d.\\ random variables $X_1,\\ldots,X_k$. We also characterize all maximizing distributions. Using this result, we prove the Erdős Matching Conjecture for every fixed $k\\ge3$, all sufficiently large $s$, and every $n\\ge k(s+1)$. More specifically, we show that in this range, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $s+1$ pairwise disjoint edges is \\[ \\max\\left\\{ \\binom{k(s+1)-1}{k}, \\binom nk-\\binom{n-s}{k} \\right\\}. \\] Moreover, every extremal family consists either of all $k$-sets meeting a fixed $s$-set or of all $k$-sets contained in a fixed $(k(s+1)-1)$-set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16203",
  "author_total": 3,
  "authors": [
    {
      "name": "Cheng Chi"
    },
    {
      "name": "Shuang Sun"
    },
    {
      "name": "Yan Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0xfn3ds5h37v6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16203",
  "fulltext_url": null,
  "id": "0xfn3ds5h37v6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16203",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7221832478",
  "pdf_url": "https://arxiv.org/pdf/2610.16203",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0xfn3ds5h37v6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16203",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16203"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A Proof of the Erdős Matching Conjecture for Large Matchings via Sharp Tail Probability for Sums of i.i.d. Nonnegative Random Variables",
  "title_text": "A Proof of the Erdős Matching Conjecture for Large Matchings via Sharp Tail Probability for Sums of i.i.d. Nonnegative Random Variables",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/0xfn3ds5h37v6",
  "venue": "arXiv",
  "versions": []
}