{
  "abstract": "Let $r\\geq3$ and $k_1\\geq k_2\\geq\\cdots\\geq k_r\\geq t$. Let $\\mathcal{F}_1,\\mathcal{F}_2,\\ldots,\\mathcal{F}_r$ be families of subspaces, of respective dimensions $k_1,k_2,\\ldots,k_r$, in an $n$-dimensional vector space over the finite field $\\mathbb{F}_q$. The $r$ families are called $r$-cross $t$-intersecting if $\\dim \\left(F_{1} \\cap F_{2} \\cap \\cdots \\cap F_{r}\\right) \\geq t$ for all $F_{i} \\in \\mathcal{F}_{i}, i = 1,2,\\dots,r$. In 2016, Frankl and Tokushige conjectured that $\\prod_{i=1}^{r}|\\mathcal{F}_i|\\leq\\prod_{i=1}^{r}{n-1\\brack k_i-1}$ for $t=1$ and $n\\geq rk_1/(r-1)$. The appealing conjecture suggests establishing intersection theorems for $n\\sim ck_1$ with $c=c(r)\\in(1,2)$, a direction that has long been challenging. In this paper, we overcome this barrier by proving that $$\\prod_{i=1}^{r}|\\mathcal{F}_i|\\leq\\prod_{i=1}^{r}{n-t\\brack k_i-t}\\;\\;\\mbox{for all}\\;\\;t\\geq1\\;\\mbox{and}\\;n\\geq rk_1/(r-1)+C(t,r),$$ where $C(t,r)=rt/(r-1)+1$. This proves the Frankl--Tokushige conjecture except for at most three values of $n$, and establishes an Erdős--Ko--Rado type theorem for almost all values of parameters. Furthermore, we characterize all extremal configurations. Our proof is purely combinatorial and based on the $t$-cover method, with several essential refinements. We also obtain almost complete intersection theorems for $r$-wise $t$-intersecting families and non-trivial $r$-cross $t$-intersecting families.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.05848",
  "author_total": 3,
  "authors": [
    {
      "name": "Yao Li"
    },
    {
      "name": "Benjian Lv"
    },
    {
      "name": "Jie Wen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ea2azbd48zant.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.05848",
  "fulltext_url": null,
  "id": "ea2azbd48zant",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.05848",
  "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": "W7212031121",
  "pdf_url": "https://arxiv.org/pdf/2609.05848",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ea2azbd48zant.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.05848",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.05848"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On the Frankl--Tokushige conjecture and almost complete $r$-cross $t$-intersection theorems for vector spaces",
  "title_text": "On the Frankl--Tokushige conjecture and almost complete r-cross t-intersection theorems for vector spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/ea2azbd48zant",
  "venue": "arXiv",
  "versions": []
}