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{
    "pk": 64902,
    "title": "On the sizes of \\(t\\)-intersecting \\(k\\)-chain-free families",
    "subtitle": null,
    "abstract": "A set system \\({\\mathcal F}\\) is \\(t\\)-intersecting, if the size of the intersection of every pair of its elements has size at least \\(t\\). A set system \\({\\mathcal F}\\) is \\(k\\)-Sperner, if it does not contain a chain of length \\(k+1\\). Our main result is the following: Suppose that \\(k\\) and \\(t\\) are fixed positive integers, where \\(n+t\\) is even and \\(n\\) is large enough. If \\({\\mathcal F}\\subseteq 2^{[n]}\\) is a \\(t\\)-intersecting \\(k\\)-Sperner family, then \\(|{\\mathcal F}|\\) has size at most the size of the sum of \\(k\\) layers, of sizes \\((n+t)/2,\\ldots, (n+t)/2+k-1\\). This bound is best possible. The case when \\(n+t\\) is odd remains open.\n \nMathematics Subject Classifications: 05D05\n \nKeywords: Extremal set theory, Sperner families, intersection theorems",
    "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": "Extremal set theory"
        },
        {
            "word": "Sperner families"
        },
        {
            "word": "intersection theorems"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/8236n14k",
    "frozenauthors": [
        {
            "first_name": "József",
            "middle_name": "",
            "last_name": "Balogh",
            "name_suffix": "",
            "institution": "University of Illinois at Urbana-Champaign, Champaign, U.S.A.",
            "department": ""
        },
        {
            "first_name": "William",
            "middle_name": "B.",
            "last_name": "Linz",
            "name_suffix": "",
            "institution": "University of South Carolina, Columbia, U.S.A.",
            "department": ""
        },
        {
            "first_name": "Balázs",
            "middle_name": "",
            "last_name": "Patkós",
            "name_suffix": "",
            "institution": "Alfréd Rényi Institute of Mathematics, Budapest, Hungary",
            "department": ""
        }
    ],
    "date_submitted": "2023-09-14T08:06:16Z",
    "date_accepted": "2023-09-14T08:06:16Z",
    "date_published": "2023-09-15T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64902/galley/49712/download/"
        }
    ]
}