A New Upper Bound for the Ramsey Number of Fans

TL;DR

This paper improves the upper bound for the Ramsey number of fan graphs, reducing it from previous estimates to a tighter bound of approximately 5.17n plus a constant, advancing understanding of these graph parameters.

Contribution

The paper introduces a new method to establish a sharper upper bound for the Ramsey number of fan graphs, improving upon prior bounds by Chen, Yu, and Zhao.

Findings

New upper bound for R(F_n) is approximately 5.17n plus a constant.

Previous bounds were 4.5n-5 and 5.5n+6, now improved to 31n/6+O(1).

The result narrows the gap in understanding Ramsey numbers for fan graphs.

Abstract

A fan $F_n$ is a graph consisting of $n$ triangles, all having precisely one common vertex. Currently, the best known bounds for the Ramsey number $R(F_n)$ are $9n/2-5 \leq R(F_n) \leq 11n/2+6$, obtained by Chen, Yu and Zhao. We improve the upper bound to $31n/6+O(1)$.