Abstract
Burr, Erdős, 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 $n \ge 25m$. 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.
Highlights
The generalized Ramsey number R(H, K) is the smallest positive integer N such for any graph G with at least N vertices either G contains H as a subgraph or its complement G contains K as a subgraph, where H and K are any two given graphs
Erdos, Faudree, Rousseau and Schelp initiated the study of Ramsey numbers of trees versus odd cycles, proving that R(Tn, Cm) = 2n − 1 for all odd m 3 and n 756m10, where Tn is a tree with n vertices and Cm is an odd cycle of length m
Generalized Ramsey numbers have since been well studied for a variety of graphs, including trees and odd cycles
Summary
The generalized Ramsey number R(H, K) is the smallest positive integer N such for any graph G with at least N vertices either G contains H as a subgraph or its complement G contains K as a subgraph, where H and K are any two given graphs. In 1982, Burr, Erdos, Faudree, Rousseau and Schelp showed that for sufficiently large n and small , Burr’s lower bound on the Ramsey numbers of sparse connected graphs G with at most (1 + )n edges versus odd cycles Cm is tight. If G is a connected graph with n vertices and at most n(1 + 1/42m5) edges where m 3 is odd and n 756m10, R(G, Cm) = 2n − 1 This theorem implies the following corollary identifying the Ramsey number of trees versus odd cycles for n very large relative to m. This extremal graph Kn−1,n−1 will be useful in motivating our proof of Theorem 4, the last steps of which are devoted to showing that any graph that is any counterexample to Theorem 4 would necessarily have a similar structure to Kn−1,n−1
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