A Tverberg-type problem of Kalai: Two negative answers to questions of Alon and Smorodinsky, and the power of disjointness

TL;DR

This work answers two key questions in Kalai's Tverberg-type framework for unions of convex sets by showing $f_r(d,s,\ldots,s) > s^r$ for all $r\ge 2$, $d\ge 2r-2$, thereby excluding a linear- or polynomial-in-$s$ regime in general. The authors construct a planar scalloped $s$-gon gadget and lift it to a high-dimensional torus to prove the lower bound, while also introducing a disjoint-union variant $F_r(d,s_1,\ldots,s_r)$ that reveals markedly different behavior, including $F_2(2,s,s)=O(s\log s)$. They connect the disjoint-union problem to hypergraph Turán theory and a variant of the Erdős box problem, yielding upper bounds across regimes and highlighting the power of disjointness in geometric intersection phenomena. Together, the results illustrate the subtleties and limitations of VC-dimension-based upper bounds and reveal rich connections between combinatorial geometry and Turán-type hypergraph theory. The methods fuse a planar-geometry gadget with a high-dimensional product construction, offering a blueprint for further explorations of Kalai-type questions and their algorithmic implications.

Abstract

Let $f_r(d,s_1,\ldots,s_r)$ denote the least integer $n$ such that every $n$-point set $P\subseteq\mathbb{R}^d$ admits a partition $P=P_1\cup\cdots\cup P_r$ with the property that for any choice of $s_i$-convex sets $C_i\supseteq P_i$ $(i\in[r])$ one necessarily has $\bigcap_{i=1}^r C_i\neq\emptyset$, where an $s_i$-convex set means a union of $s_i$ convex sets. A recent breakthrough by Alon and Smorodinsky establishes a general upper bound $f_r(d,s_1,\dots,s_r) = O(dr^2\log r \prod_{i=1}^r s_i\cdot \log(\prod_{i=1}^r s_i).$ Specializing to $r=2$ resolves the problem of Kalai from the 1970s. They further singled out two particularly intriguing questions: whether $f_{2}(2,s,s)$ can be improved from $O(s^2\log s)$ to $O(s)$, and whether $f_r(d,s,\ldots,s)\le Poly(r,d,s)$. We answer both in the negative by showing the exponential lower bound $f_{r}(d,s,\ldots,s)> s^{r}$ for any $r\ge 2$, $s\ge 1$ and $d\ge 2r-2$, which matches the upper bound up to a multiplicative $\log{s}$ factor for sufficiently large $s$. Our construction combines a scalloped planar configuration with a direct product of regular $s$-gon on the high-dimensional torus $(\mathbb{S}^1)^{r-2}$. Perhaps surprisingly, if we additionally require that within each block the $s_i$ convex sets are pairwise disjoint, the picture changes markedly. Let $F_r(d,s_1,\ldots,s_r)$ denote this disjoint-union variant of the extremal function. We show: (1) $F_{2}(2,s,s)=O(s\log s)$ by connecting it to a suitable line-separating function in the plane; (2) when $s$ is large, $F_r(d,s,\ldots,s)$ can be bounded by $O_{r,d}(s^{(1-\frac{1}{2^{d}(d+1)})r+1})$ and $O_{d}(r^{3}\log r\cdot s^{2d+3})$, respectively. This builds on a novel connection between the geometric obstruction and hypergraph Turán numbers, in particular, a variant of the Erdős box problem.

Figures (5)

• Figure 2.1: Scalloped $10$-gon
• Figure 2.2: Scalloped $30$-gon
• Figure 2.3: We select $s = 6$ points on each circular arc. This figure illustrates that each convex polygon intersects each disk only at the selected points.
• Figure 3.1: An illustration of the high-dimensional construction.
• Figure 6.1: $f_{2}(2,4,4)>32$

Theorems & Definitions (44)

• Theorem 1.1: 2025arxivAlonSmo
• Theorem 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Claim 2.2
• proof : Proof of claim
• Lemma 2.3
• proof : Proof of Lemma \ref{['lemma:KeyInConstruction']}