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{
    "pk": 64976,
    "title": "On the maximum degree of induced subgraphs of the Kneser graph",
    "subtitle": null,
    "abstract": "For integers \\(n \\geq k \\geq 1\\), the Kneser graph \\(K(n, k)\\) is the graph with vertex-set consisting of all the \\(k\\)-element subsets of \\(\\{1,2,\\ldots,n\\}\\), where two \\(k\\)-element sets are adjacent in \\(K(n,k)\\) if they are disjoint. We show that if \\((n,k,s) \\in \\mathbb{N}^3\\) with \\(n › 10000 k s^5\\) and \\(\\mathcal F\\) is set of vertices of \\(K(n,k)\\) of size larger than \\[\\{A \\subset \\{1,2,\\ldots,n\\}: |A|=k, A \\cap \\{1,2,\\ldots,s\\} \\neq \\varnothing\\},\\] then the subgraph of \\(K(n,k)\\) induced by \\(\\mathcal F\\) has maximum degree at least \\[ \\left(1 - O\\left(\\sqrt{s^3 k/n}\\right)\\right)\\frac{s}{s+1} \\cdot {n-k \\choose k} \\cdot \\frac{\\left|{\\mathcal F}\\right|}{\\binom{n}{k}}.\\] This is sharp up to the behaviour of the error term \\(O(\\sqrt{s^3 k/n})\\). In particular, if the triple of integers \\((n, k, s)\\) satisfies the condition above, then the minimum maximum degree does not increase `continuously' with \\(\\left|{\\mathcal F}\\right|\\). Instead, it has \\(s\\) jumps, one at each time when \\(\\left|{\\mathcal F}\\right|\\) becomes just larger than the union of \\(i\\) stars, for \\(i = 1, 2, \\ldots, s\\). An appealing special case of the above result is that if \\(\\mathcal{F}\\) is a family of \\(k\\)-element subsets of \\(\\{1,2,\\ldots,n\\}\\) with \\(|\\mathcal{F}| = {n-1 \\choose k-1}+1\\), then there exists \\(A \\in \\mathcal{F}\\) such that \\(\\mathcal{F}\\) is disjoint from at least \\[\\left(1/2-O\\left(\\sqrt{k/n}\\right)\\right){n-k-1 \\choose k-1}\\] of the other sets in \\(\\mathcal{F}\\); we give both a random and a deterministic construction showing that this is asymptotically sharp if \\(k=o(n)\\). In addition, it solves (up to a constant multiplicative factor) a problem of Gerbner, Lemons, Palmer, Patkós and Szécsi.\nFrankl and Kupavskii, using different methods, have recently proven similar results under the hypothesis that \\(n\\) is at least a quadratic in \\(k\\).\n \nMathematics Subject Classifications: 05D05\n \nKeywords: Kneser, intersecting, sensitivity, Erdős-Ko-Rado type theorem",
    "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": "Kneser"
        },
        {
            "word": "intersecting"
        },
        {
            "word": "sensitivity"
        },
        {
            "word": "Erdős-Ko-Rado type theorem"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/19t615fh",
    "frozenauthors": [
        {
            "first_name": "Hou",
            "middle_name": "Tin",
            "last_name": "Chau",
            "name_suffix": "",
            "institution": "School of Mathematics, University of Bristol, Bristol, U.K.",
            "department": ""
        },
        {
            "first_name": "David",
            "middle_name": "",
            "last_name": "Ellis",
            "name_suffix": "",
            "institution": "School of Mathematics, University of Bristol, Bristol, U.K.",
            "department": ""
        },
        {
            "first_name": "Ehud",
            "middle_name": "",
            "last_name": "Friedgut",
            "name_suffix": "",
            "institution": "Department of Mathematics, Weizmann Institute of Science, Rehovot, Israel.",
            "department": ""
        },
        {
            "first_name": "Noam",
            "middle_name": "",
            "last_name": "Lifshitz",
            "name_suffix": "",
            "institution": "Einstein Institute of Mathematics, Hebrew University of Jersulem, Jerusalem, Israel.",
            "department": ""
        }
    ],
    "date_submitted": "2025-03-14T20:07:01+03:00",
    "date_accepted": "2025-03-14T20:07:01+03:00",
    "date_published": "2025-03-15T10:00:00+03:00",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64976/galley/49786/download/"
        }
    ]
}