A New Upper Bound for the Ramsey Number of Fans

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 $9n\mathrm{/}2-5\le R(F_n)\le 11n\mathrm{/}2+6$, obtained by Chen, Yu and Zhao. We improve the upper bound to $31n\mathrm{/}6+O(1)$.