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{
    "pk": 64928,
    "title": "Short proofs of three results about intersecting systems",
    "subtitle": null,
    "abstract": "In this note, we give short proofs of three theorems about intersection problems. The first one is a determination of the maximum size of a nontrivial \\(k\\)-uniform, \\(d\\)-wise intersecting family for \\(n\\ge \\left(1+\\frac{d}{2}\\right)(k-d+2)\\), which improves the range of \\(n\\) of a recent result of O'Neill and Verstraëte. Our proof also extends to \\(d\\)-wise, \\(t\\)-intersecting families, and from this result we obtain a version of the Erdős-Ko-Rado theorem for \\(d\\)-wise, \\(t\\)-intersecting families.\nOur second result partially proves a conjecture of Frankl and Tokushige about \\(k\\)-uniform families with restricted pairwise intersection sizes.\nOur third result is about intersecting families of graphs. Answering a question of Ellis, we construct \\(K_{s, t}\\)-intersecting families of graphs which have size larger than the Erdős-Ko-Rado-type construction, whenever \\(t\\) is sufficiently large in terms of \\(s\\). The construction is based on nontrivial \\((2s)\\)-wise \\(t\\)-intersecting families of sets.\n \nMathematics Subject Classifications: 05D05, 05D99\n \nKeywords: Nontrivial intersecting family, Hilton-Milner, forbidden intersection, graph intersection",
    "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": "Nontrivial intersecting family"
        },
        {
            "word": "Hilton-Milner"
        },
        {
            "word": "forbidden intersection"
        },
        {
            "word": "graph intersection"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/8ph5f0p8",
    "frozenauthors": [
        {
            "first_name": "József",
            "middle_name": "",
            "last_name": "Balogh",
            "name_suffix": "",
            "institution": "Department of Mathematical Sciences, University of Illinois at Urbana-Champaign, IL, U.S.A.",
            "department": ""
        },
        {
            "first_name": "William",
            "middle_name": "",
            "last_name": "Linz",
            "name_suffix": "",
            "institution": "Current Address: Department of Mathematics, University of South Carolina, Columbia, SC, U.S.A Research performed while at: Department of Mathematical Sciences, University of Illinois at Urbana-Champaign, IL, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2024-07-01T11:27:11+02:00",
    "date_accepted": "2024-07-01T11:27:11+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/64928/galley/49738/download/"
        }
    ]
}