{
  "abstract": "Let $f_{r}(d,s_{1},\\ldots,s_{r})$ be the least $N$ such that every $N$-point set $P\\subseteq\\mathbb{R}^{d}$ has an $r$-partition $P=P_{1}\\sqcup\\cdots\\sqcup P_{r}$ with the following property: whenever $C_{i}\\supseteq P_{i}$ is a union of at most $s_{i}$ convex sets, one has $\\bigcap_{i=1}^{r}C_{i}\\ne\\emptyset$. A recent breakthrough of Alon and Smorodinsky proved that $f_{r}(d,s,\\ldots,s)\\le cdr^{2}s^{r}\\log r\\log(es^{r})$ for an absolute constant $c>0$. In this paper, we determine the asymptotic order in two principal ranges: $f_{2}(2,s,s)=Θ(s^{2})$, and $f_{r}(d,s,\\ldots,s)=Θ_{d,r}(s^{r}\\log{s})$ for every fixed $d,r$ with $d\\ge r+2$. The first one determines the order of the extremal function proposed by Kalai from the 1970s. Together, the two results show a sharp dependence on the dimension: for two parts, the logarithmic factor disappears in the plane but is necessary in every fixed dimension $d\\ge4$. Beyond these sharp results, when $r\\ge d+1$ we improve the upper bound of Alon and Smorodinsky by proving both $f_{r}(d,s,\\ldots,s)\\le c_{d}rs^{r}\\log(ers^{r})$ and $f_{r}(d,s,\\ldots,s)\\le c_{d}r^{d+2}s^{d+1}\\log(ers)$ through a local Helly-type argument. We also prove $f_{r}(d,s,\\ldots,s)>s^{\\min\\{r,d\\}}$ for every $d\\ge2$. Finally, we study two colored analogues. The direct Bárány--Larman-type extension, in which one seeks $r$ disjoint rainbow sets chosen from $d+1$ color classes, fails as soon as two convex pieces are allowed. Nevertheless, a different extension does hold: given sufficiently many prescribed $r$-point classes, one can split every class completely among the $r$ final parts while retaining the required intersection property.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.12449",
  "author_total": 3,
  "authors": [
    {
      "name": "Gennian Ge"
    },
    {
      "name": "Yang Shu"
    },
    {
      "name": "Zixiang Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aeea16t8f85ng.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.12449",
  "fulltext_url": null,
  "id": "aeea16t8f85ng",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.12449",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2607|Advanced Combinatorial Mathematics"
    ]
  ],
  "openalex": "W7168384268",
  "pdf_url": "https://arxiv.org/pdf/2607.12449",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aeea16t8f85ng.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2607.12449",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.12449"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Tverberg's theorem for unions of convex sets: Sharp bounds and colored extensions",
  "title_text": "Tverberg's theorem for unions of convex sets: Sharp bounds and colored extensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/aeea16t8f85ng",
  "venue": "arXiv",
  "versions": []
}