On the geometric Ramsey numbers of trees

TL;DR

This paper establishes upper bounds for geometric Ramsey numbers involving trees and specific graph classes, providing exact values for certain trees and broad polynomial bounds for general cases.

Contribution

It introduces new upper bounds for geometric Ramsey numbers of trees, including exact formulas for caterpillars and outerplanar graphs, and polynomial bounds for arbitrary trees.

Findings

Exact value: R_c(T_n,H_m)=(n-1)(m-1)+1 for caterpillars and Hamiltonian outerplanar graphs.

Exact value: R_g(T_n,H_m)=(n-1)(m-1)+1 if T_n has at most two non-leaf vertices.

Polynomial upper bounds: O(n^2 m) for certain cases, O(n^3 m^2) for general trees.

Abstract

In this paper, we obtain upper bounds for the geometric Ramsey numbers of trees. We prove that $R_c(T_n,H_m)=(n-1)(m-1)+1$ if $T_n$ is a caterpillar and $H_m$ is a Hamiltonian outerplanar graph on $m$ vertices. Moreover, if $T_n$ has at most two non-leaf vertices, then $R_g(T_n,H_m)=(n-1)(m-1)+1$. We also prove that $R_c(T_n,H_m)=O(n^2m)$ and $R_g(T_n,H_m)=O(n^3m^2)$ if $T_n$ is an arbitrary tree on $n$ vertices and $H_m$ is an outerplanar triangulation with pathwidth 2. %Further, we prove a uniform polynomial upper bound for the geometric Ramsey numbers of caterpillars and we also give an upper bound for $R_g(T_n)$ where $T_n$ is an arbitrary tree.