Radon numbers grow linearly

2019/12/04 by Pálvölgyi, Dömötör · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1912.02239

Abstract

Define the k-th Radon number r_k of a convexity space as the smallest number (if it exists) for which any set of r_k points can be partitioned into k parts whose convex hulls intersect. Combining the recent abstract fractional Helly theorem of Holmsen and Lee with earlier methods of Bukh, we prove that r_k grows linearly, i.e., r_k≤ c(r₂)⋅ k.

Cited by

• Strong invariants and Tverberg numbers in convexity spaces
• Extended VC-dimension, and Radon and Tverberg type theorems for unions of convex sets

Related

• Radon numbers and the fractional Helly theorem
• The fractional Helly number for separable convexity spaces
• Helly type problems in convexity spaces
• A fractional Helly theorem for set systems with slowly growing homological shatter function
• Radon partitions in convexity spaces