Radon numbers grow linearly

TL;DR

This paper proves that the k-th Radon number in a convexity space increases linearly with k, using advanced combinatorial and geometric theorems to establish this growth rate.

Contribution

It establishes a linear upper bound for Radon numbers in convexity spaces, combining recent fractional Helly theorems with classical methods.

Findings

Radon numbers grow linearly with k

Established a bound r_k ≤ c(r_2)·k

Connected fractional Helly theorem with Radon number growth

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., $r_k\le c(r_2)\cdot k$.