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{
    "pk": 64935,
    "title": "Exact antichain saturation numbers via a generalisation of a result of Lehman-Ron",
    "subtitle": null,
    "abstract": "For given positive integers \\(k\\) and \\(n\\), a family \\(\\mathcal{F}\\) of subsets of \\(\\{1,\\dots,n\\}\\) is \\(k\\)-antichain saturated if it does not contain an antichain of size \\(k\\), but adding any set to \\(\\mathcal{F}\\) creates an antichain of size \\(k\\). We use sat\\(^*(n, k)\\) to denote the smallest size of such a family. For all \\(k\\) and sufficiently large \\(n\\), we determine the exact value of sat\\(^*(n, k)\\). Our result implies that sat\\(^*(n, k)=n(k-1)-\\Theta(k\\log k)\\), which confirms several conjectures on antichain saturation. Previously, exact values for sat\\(^*(n,k)\\) were only known for \\(k\\) up to \\(6\\).\nWe also prove a strengthening of a result of Lehman-Ron which may be of independent interest. We show that given \\(m\\) disjoint chains \\(C^1,\\dots,C^m\\) in the Boolean lattice, we can create \\(m\\) disjoint skipless chains that cover the elements from \\(\\cup_{i=1}^mC^i\\) (where we call a chain skipless if any two consecutive elements differ in size by exactly one).\n \nMathematics Subject Classifications: 06A07, 05D99\n \nKeywords: Skipless chains, poset saturation, antichain saturation, Boolean lattice",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Skipless chains"
        },
        {
            "word": "poset saturation"
        },
        {
            "word": "antichain saturation"
        },
        {
            "word": "Boolean lattice"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/2s4544d9",
    "frozenauthors": [
        {
            "first_name": "Paul",
            "middle_name": "",
            "last_name": "Bastide",
            "name_suffix": "",
            "institution": "Laboratoire Bordelais de Recherche en Informatique, France -- Technische Universiteit Delft, The Netherlands",
            "department": ""
        },
        {
            "first_name": "Carla",
            "middle_name": "",
            "last_name": "Groenland",
            "name_suffix": "",
            "institution": "Technische Universiteit Delft, The Netherlands",
            "department": ""
        },
        {
            "first_name": "Hugo",
            "middle_name": "",
            "last_name": "Jacob",
            "name_suffix": "",
            "institution": "Laboratoire d'Informatique, de Robotique et de Microélectronique de Montpellier, France",
            "department": ""
        },
        {
            "first_name": "Tom",
            "middle_name": "",
            "last_name": "Johnston",
            "name_suffix": "",
            "institution": "University of Bristol and Heilbronn Institute for Mathematical Research, U.K.",
            "department": ""
        }
    ],
    "date_submitted": "2024-07-01T15:34:19+02:00",
    "date_accepted": "2024-07-01T15:34:19+02:00",
    "date_published": "2024-06-30T09:00:00+02:00",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64935/galley/49745/download/"
        }
    ]
}