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{
    "pk": 64967,
    "title": "Note on the number of antichains in generalizations of the boolean lattice",
    "subtitle": null,
    "abstract": "We give a short and self-contained argument that shows that, for any positive integers \\(t\\) and \\(n\\) with \\(t =O\\Bigl(\\frac{n}{\\log n}\\Bigr)\\), the number \\(\\alpha([t]^n)\\) of antichains of the poset \\([t]^n\\) is at most \\[{\\exp_2\\Bigl[\\Bigl(1+O\\Bigl(\\Bigl(\\frac{t\\log^3 n}{n}\\Bigr)^{1/2}\\Bigr)\\Bigr)N(t,n)\\Bigr]}\\,,\\] where \\(N(t,n)\\) is the size of a largest level of \\([t]^n\\). This, in particular, says that if \\({t \\!\\ll\\! n/\\log^3 \\! n}\\) as \\(n \\rightarrow \\infty\\), then \\(\\log\\alpha([t]^n)=(1+o(1))N(t,n)\\), giving a (partially) positive answer to a question of Moshkovitz and Shapira for \\(t, n\\) in this range. Particularly for \\(t=3\\), we prove a better upper bound: \\[\\log\\alpha([3]^n)\\le(1+4\\log 3/n)N(3,n),\\] which is the best known upper bound on the number of antichains of \\([3]^n\\).\n \nMathematics Subject Classifications: 05A16, 06A07\n \nKeywords: Boolean lattice, antichains, entropy method",
    "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": "Boolean lattice"
        },
        {
            "word": "antichains"
        },
        {
            "word": "entropy method"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/1cp4b92v",
    "frozenauthors": [
        {
            "first_name": "Jinyoung",
            "middle_name": "",
            "last_name": "Park",
            "name_suffix": "",
            "institution": "Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, U.S.A.",
            "department": ""
        },
        {
            "first_name": "Michail",
            "middle_name": "",
            "last_name": "Sarantis",
            "name_suffix": "",
            "institution": "Institute of Discrete Mathematics, Technical University of Graz, Austria",
            "department": ""
        },
        {
            "first_name": "Prasad",
            "middle_name": "",
            "last_name": "Tetali",
            "name_suffix": "",
            "institution": "School of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2025-03-14T16:31:09Z",
    "date_accepted": "2025-03-14T16:31:09Z",
    "date_published": "2025-03-15T07:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64967/galley/49777/download/"
        }
    ]
}