{
  "abstract": "For $n\\in\\mathbb N$, let $μ_n$ be the uniform measure on $2^{[n]}$. A family $\\mathcal F \\subseteq2^{[n]}$ is called $k$-AP-intersecting if $F_1\\cap F_2$ contains a nonconstant $k$-term arithmetic progression for every $F_1, F_2\\in\\mathcal F$. Writing $m_k$ for the supremum of $μ_n(\\mathcal F)$ over all $n$ and all such families $\\mathcal F$, we prove \\[ 2^{-k}\\le m_k\\le32\\cdot 2^{-k}\\qquad(k\\ge3). \\] This greatly improves a recent result of Gillott, who proved that $m_k \\leq k^{-1/3}$ for all sufficiently large $k$, and answers his question on $m_k$'s exponential decay. We also prove a similar bound for the corresponding cross-intersection problem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.26243",
  "author_total": 4,
  "authors": [
    {
      "name": "Andrey Kupavskii"
    },
    {
      "name": "Fedor Noskov"
    },
    {
      "name": "Georgy Sokolov"
    },
    {
      "name": "Nikolai Terekhov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2vc3rvcxhes7k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.26243",
  "fulltext_url": null,
  "id": "2vc3rvcxhes7k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.26243",
  "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"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7223060322",
  "pdf_url": "https://arxiv.org/pdf/2610.26243",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2vc3rvcxhes7k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.26243",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.26243"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Families that intersect in an arithmetic progression",
  "title_text": "Families that intersect in an arithmetic progression",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/2vc3rvcxhes7k",
  "venue": "arXiv",
  "versions": []
}