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{
    "pk": 64926,
    "title": "Colorful words and \\(d\\)-Tverberg complexes",
    "subtitle": null,
    "abstract": "We give a complete combinatorial characterization of weakly \\(d\\)-Tverberg complexes. These complexes record which intersection combinatorics of convex hulls necessarily arise in any sufficiently large general position point set in \\(\\mathbb R^d\\). This strengthens the concept of \\(d\\)-representable complexes, which describe intersection combinatorics that arise in at least one point set. Our characterization allows us to construct for every fixed \\(d\\) a graph that is not weakly \\(d'\\)-Tverberg for any \\({d' \\le d}\\), answering a question of De Loera, Hogan, Oliveros, and Yang.\n \nMathematics Subject Classifications: 52A35, 52C45\n \nKeywords: Tverberg's theorem, word representable, \\(d\\)-representable, nerve, general position, strong general position, fully independent",
    "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": "Tverberg's theorem"
        },
        {
            "word": "word representable"
        },
        {
            "word": "\\(d\\)-representable"
        },
        {
            "word": "nerve"
        },
        {
            "word": "general position"
        },
        {
            "word": "strong general position"
        },
        {
            "word": "fully independent"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/3ff81565",
    "frozenauthors": [
        {
            "first_name": "Florian",
            "middle_name": "",
            "last_name": "Frick",
            "name_suffix": "",
            "institution": "Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A., and Institut für Mathematik, Freie Universität Berlin, Arnimallee 2, 14195 Berlin, Germany",
            "department": ""
        },
        {
            "first_name": "R.",
            "middle_name": "Amzi",
            "last_name": "Jeffs",
            "name_suffix": "",
            "institution": "Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, U.S.A.",
            "department": ""
        }
    ],
    "date_submitted": "2024-06-28T13:45:02+02:00",
    "date_accepted": "2024-06-28T13:45:02+02:00",
    "date_published": "2024-06-30T09:00:00+02:00",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64926/galley/49736/download/"
        }
    ]
}