Ramsey numbers of trees versus odd cycles

TL;DR

This paper improves the bounds on the minimum size of trees for which the Ramsey number with odd cycles is exactly 2n-1, showing that this threshold grows linearly with the cycle length.

Contribution

The authors prove that the threshold n_0(m) for the Ramsey number R(T_n, C_m) to equal 2n-1 is at most linear in m, refining previous bounds.

Findings

Proved R(T_n, C_m) = 2n - 1 for n ≥ 50m

Established n_0(m) is bounded between two linear functions

Identified the growth rate of n_0(m) up to a constant factor

Abstract

Burr, Erd\H{o}s, Faudree, Rousseau and Schelp initiated the study of Ramsey numbers of trees versus odd cycles, proving that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 756m^{10}$, where $T_n$ is a tree with $n$ vertices and $C_m$ is an odd cycle of length $m$. They proposed to study the minimum positive integer $n_0(m)$ such that this result holds for all $n \ge n_0(m)$, as a function of $m$. In this paper, we show that $n_0(m)$ is at most linear. In particular, we prove that $R(T_n, C_m) = 2n - 1$ for all odd $m \ge 3$ and $n \ge 50m$. Combining this with a result of Faudree, Lawrence, Parsons and Schelp yields $n_0(m)$ is bounded between two linear functions, thus identifying $n_0(m)$ up to a constant factor.