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{
    "pk": 65024,
    "title": "Sidorenko hypergraphs and random Turán numbers",
    "subtitle": null,
    "abstract": "Let \\(\\mathrm{ex}(G_{n,p}^r,F)\\) denote the maximum number of edges in an \\(F\\)-free subgraph of the random \\(r\\)-uniform hypergraph \\(G_{n,p}^r\\), and let \\[s(F):=\\sup\\{s: \\exists H, t_F(H)=t_{K_r^r}(H)^{s+e(F)}›0\\}.\\] Following recent work of Conlon, Lee, and Sidorenko, we prove non-trivial lower bounds on \\(\\mathrm{ex}(G_{n,p}^r,F)\\) whenever \\(s(F)›0\\), i.e. \\(F\\) is not Sidorenko. This connection between Sidorenko's conjecture and random Turán problems gives new lower bounds on \\(\\mathrm{ex}(G_{n,p}^r,F)\\) whenever \\(s(F)›0\\), and further allows us to establish upper bounds for \\(s(F)\\) whenever upper bounds for \\(\\mathrm{ex}(G_{n,p}^r,F)\\) are known. As a consequence, we prove that \\(s(\\mathrm{E}^r(K_{k+1}^k))=\\frac{1}{r-k}\\) where \\(\\mathrm{E}^r(K_{k+1}^k)\\) is the \\(r\\)-expansion of \\(K_{k+1}^k\\).\n \nMathematics Subject Classifications: 05C65, 05C80, 05D05, 05D40\n \nKeywords: Hypergraph, Sidorenko Conjecture, Random Turán Problem",
    "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": "Hypergraph"
        },
        {
            "word": "Sidorenko Conjecture"
        },
        {
            "word": "Random Turán Problem"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/0c25t471",
    "frozenauthors": [
        {
            "first_name": "Jiaxi",
            "middle_name": "",
            "last_name": "Nie",
            "name_suffix": "",
            "institution": "School of Mathematics, Georgia Institute of Technology, Atlanta, GA, U.S.A.",
            "department": ""
        },
        {
            "first_name": "Sam",
            "middle_name": "",
            "last_name": "Spiro",
            "name_suffix": "",
            "institution": "Department of Mathematics, Rutgers University, Piscataway, NJ, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2026-02-02T10:01:58Z",
    "date_accepted": "2026-02-02T10:01:58Z",
    "date_published": "2025-12-20T08:00:00Z",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/65024/galley/49834/download/"
        }
    ]
}