{
  "abstract": "We study balanced subfamilies of the middle layer $\\binom{[2n]}{n}$ of the Boolean lattice $2^{[2n]}$. A family $\\mathcal{F}\\subseteq\\binom{[2n]}{n}$ is said to be balanced if every element in $[2n]$ appears in the same number of members of $\\mathcal{F}$. A balanced subfamily of size 2 is exactly a complementary pair $\\{A,[2n]\\setminus A\\}$, and therefore a family with no balanced subfamily of size $2$ has at most $\\frac{1}{2}\\binom{2n}{n}$ members. We show that for every $k\\geq 1$ and all sufficiently large $n$, this maximum size is compatible with delaying the smallest size of a balanced subfamily until $2k+2$. More precisely, there exists a family $\\mathcal{F}\\subseteq\\binom{[2n]}{n}$ of size $\\frac{1}{2}\\binom{2n}{n}$ with no balanced subfamilies of sizes $2,4,\\ldots,2k$, but with a balanced subfamily of size $2k+2$. The proof is constructive and uses a general lifting from simple games to self-dual selectors in the middle layer. This provides an efficient constructive proof of a recent conjecture of Moss and Pedersen.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.16172",
  "author_total": 1,
  "authors": [
    {
      "name": "Sa'ul A. Blanco"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b0wp85jv5q7rz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.16172",
  "fulltext_url": null,
  "id": "b0wp85jv5q7rz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.16172",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7164918738",
  "pdf_url": "https://arxiv.org/pdf/2606.16172",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b0wp85jv5q7rz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2606.16172",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.16172"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On balanced subfamilies of maximum complement-free families in the middle layer of the Boolean lattice",
  "title_text": "On balanced subfamilies of maximum complement-free families in the middle layer of the Boolean lattice",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/b0wp85jv5q7rz",
  "venue": "arXiv",
  "versions": []
}