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{
    "pk": 65036,
    "title": "Colorful intersections and Tverberg partitions",
    "subtitle": null,
    "abstract": "We prove an extension of Tverberg's classical result on partitioning a point set in \\(\\mathbb{R}^d\\) by replacing the point set with families of convex sets which satisfy the colorful Helly hypothesis. In particular, we show the following for any integers \\(d \\geq m \\geq 1\\) and \\(r\\) a prime power. Suppose \\(F_1, F_2, \\dots, F_m\\) are families of convex sets in \\(\\mathbb{R}^d\\), each of size \\({n › (\\frac{d}{m}+1)(r-1)}\\), such that for every choice \\(C_1\\in F_1, C_2\\in F_2, \\dots,C_m \\in F_m\\) we have \\(\\bigcap_{i=1}^mC_i\\neq \\varnothing\\). Then, one of the families \\(F_i\\) admits a Tverberg \\(r\\)-partition. That is, one of the families \\(F_i\\) can be partitioned into \\(r\\) nonempty parts such that the convex hulls of the parts have nonempty intersection. As a corollary, we extend the work of Karasev and Montejano concerning geometric transversals to families of convex sets in \\(\\mathbb{R}^d\\) that satisfy the colorful Helly hypothesis.\n \nMathematics Subject Classifications: 52A35, 57Q70\n \nKeywords: Tverberg's theorem, geometric transversals, topological combinatorics, configuration space/test map, discrete Morse theory",
    "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": "geometric transversals"
        },
        {
            "word": "topological combinatorics"
        },
        {
            "word": "configuration space/test map"
        },
        {
            "word": "discrete Morse theory"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/3z8130s5",
    "frozenauthors": [
        {
            "first_name": "Michael",
            "middle_name": "Gene",
            "last_name": "Dobbins",
            "name_suffix": "",
            "institution": "Department of Mathematics and Statistics, Binghamton University, Binghamton, New York, U.S.A.",
            "department": ""
        },
        {
            "first_name": "Andreas",
            "middle_name": "F.",
            "last_name": "Holmsen",
            "name_suffix": "",
            "institution": "Department of Mathematical Sciences, KAIST, Daejeon, South Korea -- Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea.",
            "department": ""
        },
        {
            "first_name": "Dohyeon",
            "middle_name": "",
            "last_name": "Lee",
            "name_suffix": "",
            "institution": "Department of Mathematical Sciences, KAIST, Daejeon, South Korea -- Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea.",
            "department": ""
        }
    ],
    "date_submitted": "2026-04-20T22:17:28+03:00",
    "date_accepted": "2026-04-20T22:17:28+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/65036/galley/49846/download/"
        }
    ]
}