Multicolor bipartite Ramsey number of double stars

For positive integers $n, m$, the double star $S(n,m)$ is the graph consisting of the disjoint union of two stars $K_{1,n}$ and $K_{1,m}$ together with an edge joining their centers. Finding monochromatic copies of double stars in edge-colored complete bipartite graphs has attracted much attention. The $k$-color bipartite Ramsey number of $S(n,m)$, denoted by $r_{bip}(S(n,m);k)$, is the smallest integer $N$ such that, in any $k$-coloring of the edges of the complete bipartite graph $K_{N,N}$, there is a monochromatic copy of $S(n,m)$. The study of bipartite Ramsey numbers was initiated in the early 1970s by Faudree and Schelp and, independently, by Gy\'arf\'as and Lehel. The exact value of $r_{bip}(S(n,m);k)$ is only known when $n=m=1$. Applying the Tur\'an argument in the bipartite setting, here we prove that if $k=2$ and $n\ge m$, or $k\ge3$ and $n\ge 2m$, then $$r_{bip}(S(n,m);k)=kn+1.$$