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

2025-10-23

AI Summary

This paper addresses two central conjectures on Tverberg-type problems: (1) whether $f_2(2,s,s)$ admits a linear upper bound $O(s)$, and (2) whether $f_r(d,s,dots,s)$ is polynomially bounded in $r$, $d$, and $s$. Leveraging exponential counterexamples constructed from shell-like planar configurations and direct products with high-dimensional toroidal regular polygons, the authors prove $f_r(d,s,dots,s) > s^r$, thereby refuting both conjectures. They further uncover a deep connection between these Tverberg-type functions and hypergraph Turán numbers. Under disjointness constraints, they establish tight bounds: $F_2(2,s,s) = O(s log s)$ in the plane and subexponential upper bounds in higher dimensions. These results resolve a key problem posed by Alon and Smorodinsky, revealing the intrinsic geometric complexity underlying Tverberg-type obstructions.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

Let $f_r(d,s_1,ldots,s_r)$ denote the least integer $n$ such that every $n$-point set $Psubseteqmathbb{R}^d$ admits a partition $P=P_1cupcdotscup P_r$ with the property that for any choice of $s_i$-convex sets $C_isupseteq P_i$ $(iin[r])$ one necessarily has $igcap_{i=1}^r C_i eqemptyset$, 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 $f_r(d,s_1,dots,s_r) = O(dr^2log r prod_{i=1}^r s_icdot log(prod_{i=1}^r s_i).$ 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 $O(s^2log s)$ 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 $rge 2$, $sge 1$ and $dge 2r-2$, 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) $F_{2}(2,s,s)=O(slog s)$ by performing controlled planar geometric transformations and constructing an auxiliary graph whose planarity yields the upper bound; (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 $O_{d}(r^{3}log rcdot s^{2d+3})$, respectively. This builds on a novel connection between the geometric obstruction and hypergraph Tur'{a}n numbers, in particular, a variant of the ErdH{o}s box problem.

Problem

• Determining the minimal size of point sets for Tverberg-type partitions
• Investigating intersection properties of convex set unions containing partitions
• Analyzing how disjointness requirements affect geometric partitioning bounds

Innovation

• Used scalloped planar configuration for construction
• Applied direct product of regular s-gon
• Connected geometric obstruction to hypergraph Turán numbers