A note on infinite versions of (p,q)-theorems

2024/12/05 by Attila Jung, Jung, Attila, Dömötör Pálvölgyi +1 · 1 citation
Mathematics · #52A35 #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2412.04066

openalex publication_date 2024/12/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We prove that fractional Helly and (p,q)-theorems imply (ℵ₀,q)-theorems in an entirely abstract setting. We give a plethora of applications, including reproving almost all earlier (ℵ₀,q)-theorems about geometric hypergraphs that were proved recently. Some of the corollaries are new results, for example, we prove that if F is an infinite family of convex compact sets in ℝ^d and among every ℵ₀ of the sets some d+1 contain a point in their intersection with integer coordinates, then all the members of F can be hit with finitely many points with integer coordinates.

Cited by

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

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