Tverberg's theorem for unions of convex sets: Sharp bounds and colored extensions
Abstract
Let fr(d,s1,…,sr) be the least N such that every N-point set P⊆ℝd has an r-partition P=P1sqcup⋯sqcup Pr with the following property: whenever Ci⊇ Pi is a union of at most si convex sets, one has ⋂i=1rCi≠∅. A recent breakthrough of Alon and Smorodinsky proved that fr(d,s,…,s)≤ cdr2srlog rlog(esr) for an absolute constant c>0. In this paper, we determine the asymptotic order in two principal ranges: f2(2,s,s)=Θ(s2), and fr(d,s,…,s)=Θd,r(srlogs) for every fixed d,r with d≥ 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≥4. Beyond these sharp results, when r≥ d+1 we improve the upper bound of Alon and Smorodinsky by proving both fr(d,s,…,s)≤ cdrsrlog(ersr) and fr(d,s,…,s)≤ cdrd+2sd+1log(ers) through a local Helly-type argument. We also prove fr(d,s,…,s)>smin{r,d} for every d≥2. 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.
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BibTeX
@misc{ge2026tverbergs,
title = {{Tverberg's theorem for unions of convex sets: Sharp bounds and colored extensions}},
author = {Gennian Ge and Yang Shu and Zixiang Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.12449},
eprint = {2607.12449},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.12449},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Tverberg's theorem for unions of convex sets: Sharp bounds and colored extensions
AU - Gennian Ge
AU - Yang Shu
AU - Zixiang Xu
PY - 2026
DA - 2026/09/05
PB - arXiv
DO - 10.48550/arxiv.2607.12449
UR - https://arxiv.org/abs/2607.12449
AB - Let f_r(d,s_1,…,s_r) be the least N such that every N-point set P⊆ℝ^d has an r-partition P=P_1sqcup⋯sqcup P_r with the following property: whenever C_i⊇ P_i is a union of at most s_i convex sets, one has ⋂_i=1^rC_i≠∅. A recent breakthrough of Alon and Smorodinsky proved that f_r(d,s,…,s)≤ cdr^2s^rlog rlog(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,…,s)=Θ_d,r(s^rlogs) for every fixed d,r with d≥ 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≥4. Beyond these sharp results, when r≥ d+1 we improve the upper bound of Alon and Smorodinsky by proving both f_r(d,s,…,s)≤ c_drs^rlog(ers^r) and f_r(d,s,…,s)≤ c_dr^d+2s^d+1log(ers) through a local Helly-type argument. We also prove f_r(d,s,…,s)>s^min{r,d} for every d≥2. 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.
ER -