Improved bounds on the Ramsey number of fans

Guantao Chen; Xiaowei Yu; Yi Zhao

doi:10.1016/j.ejc.2021.103347

paper

Improved bounds on the Ramsey number of fans

arXiv:2007.00152 · doi:10.1016/j.ejc.2021.103347

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 $9 n / 2 - 5 \leq r (F_{n}) \leq 11 n / 2 + 6$ for any $n$ . This improves previous best bounds $r (F_{n}) \leq 6 n$ of Lin and Li and $r (F_{n}) \geq 4 n + 2$ of Zhang, Broersma and Chen.