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{
    "pk": 65033,
    "title": "Sperner systems with restricted differences",
    "subtitle": null,
    "abstract": "Let \\(\\mathcal{F}\\) be a family of subsets of \\([n]\\) and \\(L\\) be a subset of \\([n]\\). We say \\(\\mathcal{F}\\) is an \\(L\\)-differencing Sperner system if \\(|A\\setminus B|\\in L\\) for any distinct \\(A,B\\in\\mathcal{F}\\). Let \\(p\\) be a prime and \\(q\\) be a power of \\(p\\). Frankl first studied \\(p\\)-modular \\(L\\)-differencing Sperner systems and showed an upper bound of the form \\(\\sum_{i=0}^{|L|}\\binom{n}{i}\\). In this paper, we obtain new upper bounds on \\(q\\)-modular \\(L\\)-differencing Sperner systems using elementary \\(p\\)-adic analysis and polynomial method, extending and improving existing results substantially. Moreover, our techniques can be used to derive new upper bounds on subsets of the hypercube with restricted Hamming distances. One highlight of the paper is the first analogue of the celebrated Snevily's theorem in the \\(q\\)-modular setting, which results in several new upper bounds on \\(q\\)-modular \\(L\\)-avoiding \\(L\\)-intersecting systems. In particular, we improve a result of Felszeghy, Heged\\H{u}s, and Rónyai, and give a partial answer to a question posed by Babai, Frankl, Kutin, and \\v{S}tefankovič.\n \nMathematics Subject Classifications: 05D05, 11B75\n \nKeywords: Sperner theorem, separating polynomial, intersecting family, Hamming distance",
    "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": "Sperner theorem"
        },
        {
            "word": "separating polynomial"
        },
        {
            "word": "intersecting family"
        },
        {
            "word": "Hamming distance"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/65w7z20v",
    "frozenauthors": [
        {
            "first_name": "Zixiang",
            "middle_name": "",
            "last_name": "Xu",
            "name_suffix": "",
            "institution": "Extremal Combinatorics and Probability Group, Institute for Basic Science, Daejeon, South Korea",
            "department": ""
        },
        {
            "first_name": "Chi",
            "middle_name": "Hoi",
            "last_name": "Yip",
            "name_suffix": "",
            "institution": "School of Mathematics, Georgia Institute of Technology, Atlanta, GA, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2026-04-20T22:02:31+03:00",
    "date_accepted": "2026-04-20T22:02:31+03:00",
    "date_published": "2026-04-20T10:00:00+03:00",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/65033/galley/49843/download/"
        }
    ]
}