Improved bounds on the Ramsey number of fans

TL;DR

This paper establishes improved bounds on the Ramsey number of fans, a specific graph structure, narrowing the gap between previous estimates and advancing understanding of graph coloring properties.

Contribution

The paper provides tighter bounds on the Ramsey number of fan graphs, improving upon previous results and contributing to graph theory and combinatorics.

Findings

Lower bound: (9n/2)-5 for r(F_n)

Upper bound: (11n/2)+6 for r(F_n)

Bounds are tighter than previous estimates

Abstract

For a given graph $H$, the Ramsey number $r(H)$ is the minimum $N$ such that any 2-edge-coloring of the complete graph $K_N$ yields a monochromatic copy of $H$. Given a positive integer $n$, a \emph{fan }$F_n$ is a graph formed by $n$ triangles that share one common vertex. We show that ${9n}/{2}-5\le r(F_n)\le {11n}/{2} + 6$ for any $n$. This improves previous best bounds $r(F_n) \le 6n$ of Lin and Li and $r(F_n) \ge 4n+2$ of Zhang, Broersma and Chen.