{
  "abstract": "Let $M=(E,\\mathcal{I})$ be a matroid of rank $r$. Let $\\mathcal{I}_k$ be the independent sets of size $k$, and let $I_k=|\\mathcal{I}_k|$ and $I=|\\mathcal{I}|$. We show that if every set $F\\in \\mathcal{I}_{r-1}$ is contained in at least $δ$ bases, then $$\\ln \\left(\\frac{I}{I_r}\\right)\\geqslant \\frac{I_{r-1}}{I_r}\\cdot δ\\ln \\left(1+\\frac{1}δ\\right).$$ In particular, we have $$\\frac{I}{I_r}\\geqslant 2^{I_{r-1}/I_r}.$$ By combining this result with several other ideas, we prove that if $G$ is a simple connected graph on $n$ vertices, and $F(G)$ and $T(G)$ denote its numbers of spanning forests and spanning trees, respectively, then $$\\frac{F(G)}{T(G)}\\geqslant \\frac{F(K_n)}{T(K_n)},$$ where $K_n$ is the complete graph on $n$ vertices. Equality holds if and only if $G=K_n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18611",
  "author_total": 2,
  "authors": [
    {
      "name": "Ferenc Bencs"
    },
    {
      "name": "Péter Csikvári"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dv1j33nrzvt81.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18611",
  "fulltext_url": null,
  "id": "dv1j33nrzvt81",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18611",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7213503527",
  "pdf_url": "https://arxiv.org/pdf/2609.18611",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dv1j33nrzvt81.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18611",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18611"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "An inequality for the number of independent sets of matroids with an application to the forest-tree ratio of graphs",
  "title_text": "An inequality for the number of independent sets of matroids with an application to the forest-tree ratio of graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/dv1j33nrzvt81",
  "venue": "arXiv",
  "versions": []
}