Ramsey numbers of large even cycles and fans

TL;DR

This paper determines the asymptotic values of Ramsey numbers for large even cycles and fans, revealing how these numbers grow with the size of the graphs involved.

Contribution

It establishes the asymptotic behavior of Ramsey numbers for large even cycles and fans, providing explicit formulas for different ranges of the parameter a.

Findings

For large n, R(C_{2⌊an⌋}, F_n) is approximately (2+2a)n when 1/2 ≤ a < 1.

For a ≥ 1, R(C_{2⌊an⌋}, F_n) is approximately 4a n.

The results give precise growth rates of these Ramsey numbers for large graphs.

Abstract

For graphs $F$ and $H$, the Ramsey number $R(F, H)$ is the smallest positive integer $N$ such that any red/blue edge coloring of $K_N$ contains either a red $F$ or a blue $H$. Let $C_n$ be a cycle of length $n$ and $F_n$ be a fan consisting of $n$ triangles all sharing a common vertex. In this paper, we prove that for all sufficiently large $n$, \[ R(C_{2\lfloor an\rfloor}, F_n)= \left\{ \begin{array}{ll} (2+2a+o(1))n & \textrm{if $1/2\leq a< 1$,}\\ (4a+o(1))n & \textrm{if $ a\geq 1$.} \end{array} \right. \]