Radon Numbers for Trees

TL;DR

This paper explores Radon numbers within $P_3$-convexity on trees, establishing a Tverberg-type theorem and relating Radon numbers to maximal free sets, thus extending convexity concepts from Euclidean spaces to graph theory.

Contribution

It introduces Radon numbers for $P_3$-convexity in trees and proves a Tverberg-type theorem, connecting Radon numbers with maximal free sets in this convexity space.

Findings

Established a Tverberg-type theorem for $P_3$-convexity in trees.

Proved an inequality linking Radon numbers to maximal free sets.

Extended classical convexity results to graph convexity spaces.

Abstract

Many interesting problems are obtained by attempting to generalize classical results on convexity in Euclidean spaces to other convexity spaces, in particular to convexity spaces on graphs. In this paper we consider $P_3$-convexity on graphs. A set $U$ of vertices in a graph $G$ is $P_3$-convex if every vertex not in $U$ has at most one neighbour in $U$. More specifically, we consider Radon numbers for $P_3$-convexity in trees. Tverberg's theorem states that every set of $(k-1)(d+1)-1$ points in $\mathbb{R}^d$ can be partitioned into $k$ sets with intersecting convex hulls. As a special case of Eckhoff's conjecture, we show that a similar result holds for $P_3$-convexity in trees. A set $U$ of vertices in a graph $G$ is called free, if no vertex of $G$ has more than one neighbour in $U$. We prove an inequality relating the Radon number for $P_3$-convexity in trees with the size of a maximal free set.