On the multicolor Ramsey numbers of balanced double stars

The balanced double star on $2n+2$ vertices, denoted $S_{n,n}$, is the tree obtained by joining the centers of two disjoint stars each having $n$ leaves. Let $R_r(G)$ be the smallest integer $N$ such that in every $r$-coloring of the edges of $K_N$ there is a monochromatic copy of $G$, and let $R_r^{\mathrm{b}\mathrm{i}\mathrm{p}}(G)$ be the smallest integer $N$ such that in every $r$-coloring of the edges of $K_{N,N}$ there is a monochromatic copy of $G$. It is known that $R_2(S_{n,n})=3n+2$ and $R_2^{\mathrm{b}\mathrm{i}\mathrm{p}}(S_{n,n})=2n+1$ \cite{HJ}, but very little is known about $R_r(S_{n,n})$ and $R_r^{\mathrm{b}\mathrm{i}\mathrm{p}}(S_{n,n})$ when $r\ge 3$ (other than the bounds which follow from considerations on the number of edges in the majority color class). In this paper we prove the following for all $n\ge 1$ (where the lower bounds are adapted from existing examples): $$(r-1)2n+1\le R_r(S_{n,n})\le (r-\frac{1}{2})(2n+2)-1,$$and $$(2r-4)n+1\le R_r^{\mathrm{b}\mathrm{i}\mathrm{p}}(S_{n,n})\le (2r-3+\frac{2}{r}+O(\frac{1}{r^2}))n\mathrm{.}$$ These bounds are similar to the best known bounds on $R_r(P_{2n+2})$ and $R_r^{\mathrm{b}\mathrm{i}\mathrm{p}}(P_{2n+2})$, where $P_{2n+2}$ is a path on $2n+2$ vertices (which is also a balanced tree). We also give an example which improves the lower bound on $R_r^{\mathrm{b}\mathrm{i}\mathrm{p}}(S_{n,n})$ when $r=3$ and $r=5$.