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{
    "pk": 64872,
    "title": "Rainbow version of the Erdős Matching Conjecture via concentration",
    "subtitle": null,
    "abstract": "We say that the families \\(\\mathcal{F}_1,\\ldots, \\mathcal{F}_{s+1}\\) of \\(k\\)-element subsets of \\([n]\\) are cross-dependent if there are no pairwise disjoint sets \\(F_1,\\ldots, F_{s+1}\\), where \\(F_i\\in \\mathcal{F}_i\\) for each \\(i\\). The rainbow version of the Erdős Matching Conjecture due to Aharoni and Howard and independently to Huang, Loh and Sudakov states that \\(\\min_{i} |\\mathcal{F}_i|\\le \\max\\big\\{{n\\choose k}-{n-s\\choose k}, {(s+1)k-1\\choose k}\\big\\}\\) for \\(n\\ge (s+1)k\\). In this paper, we prove this conjecture for \\(n›3e(s+1)k\\) and \\(s›10^7\\). One of the main tools in the proof is a concentration inequality due to Frankl and Kupavskii.\n \nMathematics Subject Classifications: 05D05\n \nKeywords: Extremal set theory, Erdos matching conjecture, rainbow version",
    "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": "Erdos matching conjecture"
        },
        {
            "word": "rainbow version"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/2c6553mf",
    "frozenauthors": [
        {
            "first_name": "Andrey",
            "middle_name": "",
            "last_name": "Kupavskii",
            "name_suffix": "",
            "institution": "G-SCOP, CNRS, University Grenoble-Alpes, France & Moscow Institute of Physics and Technology, Russia",
            "department": ""
        }
    ],
    "date_submitted": "2023-03-14T14:14:03Z",
    "date_accepted": "2023-03-14T14:14:03Z",
    "date_published": "2023-03-15T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64872/galley/49682/download/"
        }
    ]
}