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{
    "pk": 64799,
    "title": "Families with no perfect matchings",
    "subtitle": null,
    "abstract": "We consider families of $k$-subsets of $\\{1, \\dots, n\\}$, where $n$ is a multiple of $k$, which have no perfect matching. An equivalent condition for a family $\\mathcal{F}$ to have no perfect matching is for there to be a blocking set, which is a set of $b$ elements of $\\{1, \\dots, n\\}$ that cannot be covered by $b$ disjoint sets in $\\mathcal{F}$. We are specifically interested in the largest possible size of a family $\\mathcal{F}$ with no perfect matching and no blocking set of size less than $b$. Frankl resolved the case of families with no singleton blocking set (in other words, the $b=2$ case) for sufficiently large $n$ and conjectured an optimal construction for general $b$. Though Frankl's construction fails to be optimal for $k = 2, 3$, we show that the construction is optimal whenever $k \\ge 100$ and $n$ is sufficiently large.\nMathematics Subject Classifications: 05D05",
    "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": [],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/1vc250fj",
    "frozenauthors": [
        {
            "first_name": "Mihir",
            "middle_name": "",
            "last_name": "Singhal",
            "name_suffix": "",
            "institution": "Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2021-11-11T16:12:14Z",
    "date_accepted": "2021-11-11T16:12:14Z",
    "date_published": "2021-12-15T08:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64799/galley/49609/download/"
        }
    ]
}