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

TL;DR

This paper investigates a Tverberg-type problem related to convex set intersections, providing exponential lower bounds for certain functions and exploring variants with disjoint convex sets, connecting geometric and hypergraph Turán problems.

Contribution

It establishes exponential lower bounds for the extremal functions in a Tverberg-type problem and introduces new bounds for the disjoint-union variants, linking geometry with hypergraph Turán numbers.

Findings

Exponential lower bounds for $f_r(d,s,\

disproving conjectures about polynomial bounds.

New bounds for the disjoint-union variant $F_r(d,s_1,\

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 Tur\'{a}n numbers, in particular, a variant of the Erd\H{o}s box problem.