No Infinite (p,q)-Theorem for Piercing Compact Convex Sets with Lines in ℝ³

2025/09/08 by Chakraborty, Sutanoya, Ghosh, Arijit
#Combinatorics (math.CO) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.06731

Abstract

An infinite (p,q)-theorem, or an (ℵ₀,q)-theorem, involving two families F and G of sets, states that if in every infinite subset of F, there are q sets that are intersected by some set in G, then there is a finite set S_F\subseteqG such that for every C\inF, there is a B∈ S_F with C∩ B≠∅. We provide an example demonstrating that there is no (ℵ₀,q)-theorem for piercing compact convex sets in ℝ³ with lines by constructing a family F of compact convex sets such that it does not have a finite line transversal, but for any t∈ℕ, every infinite subset of F contains t sets that are pierced by a line.

Citations

• A note on infinite versions of (p,q)-theorems
• An (ℵ₀,k+2)-Theorem for k-Transversals
• Finite k-Transversals of Infinite Families of Fat Convex Sets

Related

• An (ℵ₀,k+2)-Theorem for k-Transversals
• Finite k-Transversals of Infinite Families of Fat Convex Sets
• A geometric proof of the infinite (p, q)-theorem for hyperplane piercing
• Bounds on piercing and line-piercing numbers in families of convex sets in the plane
• Transversals to colorful intersecting convex sets