On the Ramsey Numbers of Trees with Small Diameter

TL;DR

This paper determines the Ramsey numbers for many trees of diameter 3 and shows they are all Ramsey unsaturated, advancing understanding of tree structures in Ramsey theory.

Contribution

It provides exact Ramsey numbers for numerous diameter-3 trees and proves their Ramsey unsaturation, a new property in this context.

Findings

Exact Ramsey numbers for many diameter-3 trees.

All diameter-3 trees are Ramsey unsaturated.

Enhanced understanding of tree structures in Ramsey theory.

Abstract

We estimate the Ramsey number r(T) = r(T,T) for various trees T, obtaining a precise value for r(T) for a large number of trees of diameter 3. Furthermore we prove that all trees of diameter 3 are Ramsey unsaturated as defined by Balister, Lehel, and Schelp in their article "Ramsey unsaturated and saturated graphs."