  For a graph $H$ and integer $k\ge1$, let $r(H;k)$ and $r_\ell(H;k)$ denote
the $k$-color Ramsey number and list Ramsey number of $H$, respectively.
Motivated by the work of Alon, Buci\'c, Kalvari, Kuperwasser and Szab\'o, who
initiated the systematic study of list Ramsey numbers of graphs and
hypergraphs, and conjectured that $ r(K_{1,n};k)$ and $r_\ell(K_{1,n};k)$ are
always equal, we study the $k$-color Ramsey number for double stars $S(n,m)$,
where $n\ge m\ge1$. To the best of our knowledge, little is known on the exact
value of $r(S(n,m);k)$ when $k\ge3$. A classic result of Erd\H{o}s and Graham
from 1975 asserts that $r(T;k)>k(n-1)+1$ for every tree $T$ with $n\ge 1$ edges
and $k$ sufficiently large such that $n$ divides $k-1$. Using a folklore double
counting argument in set system and the edge chromatic number of complete
graphs, we prove that if $k$ is odd and $n$ is sufficiently large compared with
$m$ and $k$, then \[ r(S(n,m);k)=kn+m+2.\] This is a step in our effort to
determine whether $r(S(n,m);k)$ and $r_\ell(S(n,m);k)$ are always equal, which
remains wide open. We also prove that $ r(S^m_n;k)=k(n-1)+m+2$ if $k $ is odd
and $n$ is sufficiently large compared with $m$ and $k$, where $1\le m\le n$
and $S^m_n$ is obtained from $K_{1, n}$ by subdividing $m$ edges each exactly
once. We end the paper with some observations towards the list Ramsey number
for $S(n,m)$ and $S^m_n$.
