Ramsey numbers of bounded degree trees versus general graphs

TL;DR

This paper establishes a precise formula for the Ramsey number involving bounded degree trees and k-chromatic graphs, confirming a conjecture and showing tight bounds up to a constant.

Contribution

It proves a tight bound for Ramsey numbers of bounded degree trees versus k-chromatic graphs, confirming a conjecture and extending understanding of graph Ramsey theory.

Findings

The Ramsey number R(T,H) equals (k-1)(|T|-1)+σ(H) for large trees.

The result is tight up to a constant factor C_{Δ,k}.

The paper confirms a conjecture by Balla, Pokrovskiy, and Sudakov.

Abstract

For every $k\ge 2$ and $\Delta$, we prove that there exists a constant $C_{\Delta,k}$ such that the following holds. For every graph $H$ with $\chi(H)=k$ and every tree with at least $C_{\Delta,k}|H|$ vertices and maximum degree at most $\Delta$, the Ramsey number $R(T,H)$ is $(k-1)(|T|-1)+\sigma(H)$, where $\sigma(H)$ is the size of a smallest colour class across all proper $k$-colourings of $H$. This is tight up to the value of $C_{\Delta,k}$, and confirms a conjecture of Balla, Pokrovskiy, and Sudakov.