On the Ramsey number of the double star

The double star $S(m_1,m_2)$ is obtained from joining the centres of a star with $m_1$ leaves and a star with $m_2$ leaves. We give a short proof of a new upper bound on the two-colour Ramsey number of $S(m_1,m_2)$, for positive $m_1,m_2$ fulfilling $(\sqrt 5+1)m_2/2 < m_1 < 3m_2$. Our result implies that for all positive $m$, the Ramsey number of the double star $S(2m,m)$ is at most $4.275m$.