{
  "abstract": "Let M = {n 1 /x 1 , . . . , n k /x k } be a multiset with arbitrary (non-uniform) multiplicities. We determine the largest pairwise-intersecting family of submultisets in two settings. Over all ranks, the maximum is the sum of max w(S), w(S c ) over complementary pairs of supports, where w(S) = i∈S n i ; it is attained by a \"weighted majority\" family, and the natural family of all submultisets containing the most frequent element is optimal if and only if\n\nAt fixed rank j, the maximum is max F S∈F c j (S) over the maximal intersecting families F of subsets of {1, . . . , k}, where c j (S) counts rank-j submultisets with support exactly S; a dominance argument reduces the 2, 4, 12, 81 families for k = 2, . . . , 5 to 1, 2, 3, 7 undominated ones, giving exact formulas for k ≤ 5. We then study the set F (M ) of ranks at which the star (all submultisets containing the most frequent element) is not optimal. With R\n\nR, and that R -1 ∈ F (M ) whenever n 3 ≥ 2; for k = 3, 4 we determine F (M ) completely (it is an interval). Computer evidence for k = 5, 6 supports the conjecture that F (M ) is always an interval. A final section gives an explicit symmetric chain decomposition of the poset of submultisets.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "name": "Sunitha K G"
    },
    {
      "name": "Girish K P"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/be771z200fqy5.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "be771z200fqy5",
  "issns": [],
  "landing_url": "https://hal.science/hal-05846994v1",
  "language": "en",
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05846994/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/be771z200fqy5.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:20:17Z",
      "native_id": "hal-05846994",
      "source": "hal",
      "url": "https://hal.science/hal-05846994v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Maximum Intersecting Families of Submultisets: A Complete Solution over All Ranks and Exact Results at Fixed Rank",
  "title_text": "Maximum Intersecting Families of Submultisets: A Complete Solution over All Ranks and Exact Results at Fixed Rank",
  "type": "preprint",
  "updated_at": "2026-11-03T12:20:17Z",
  "url": "https://research.draconitic.com/paper/be771z200fqy5",
  "venue": "HAL",
  "versions": []
}