{
  "abstract": "Motivated by the Merino--Welsh conjecture, we consider the smallest $c\\ge0$, denoted by $c_*$, for which the inequality $T_M(c,0)T_M(0,c)\\ge T_M(1,1)^2$ holds for every loopless and coloopless finite matroid $M$. The counterexamples constructed by Beke, Csáji, Csikvári, and Pituk [\\emph{Adv. Math.} \\textbf{446} (2024), 109674] give the lower bound $x_0$, where $x_0\\approx2.22668$ is the largest real root of the polynomial $x^3-9(x-1)$. Later, Csikvári [\\emph{European J. Combin.} \\textbf{137} (2026), 104402] improved the known upper bound for this constant to $2.35$ and then conjectured that the above inequality holds at $c=x_0$. This conjecture was recently proved by Liu (2026). We give an alternative direct inductive proof that $c_*=x_0$, without computer-assisted finite verification.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06589",
  "author_total": 2,
  "authors": [
    {
      "name": "Jungang Chen"
    },
    {
      "name": "Jiaxin Xie"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a63w72372f46r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06589",
  "fulltext_url": null,
  "id": "a63w72372f46r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06589",
  "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|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7220729799",
  "pdf_url": "https://arxiv.org/pdf/2610.06589",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a63w72372f46r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.06589",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06589"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A direct inductive proof of the sharp Merino--Welsh threshold for matroids",
  "title_text": "A direct inductive proof of the sharp Merino--Welsh threshold for matroids",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/a63w72372f46r",
  "venue": "arXiv",
  "versions": []
}