  Suppose that $T$ is an acyclic $r$-uniform hypergraph, with $r\ge 2$. We
define the ($t$-color) chromatic Ramsey number $\chi(T,t)$ as the smallest $m$
with the following property: if the edges of any $m$-chromatic $r$-uniform
hypergraph are colored with $t$ colors in any manner, there is a monochromatic
copy of $T$. We observe that $\chi(T,t)$ is well defined and $$\left\lceil
{R^r(T,t)-1\over r-1}\right \rceil +1 \le \chi(T,t)\le |E(T)|^t+1$$ where
$R^r(T,t)$ is the $t$-color Ramsey number of $H$. We give linear upper bounds
for $\chi(T,t)$ when T is a matching or star, proving that for $r\ge 2, k\ge 1,
t\ge 1$, $\chi(M_k^r,t)\le (t-1)(k-1)+2k$ and $\chi(S_k^r,t)\le t(k-1)+2$ where
$M_k^r$ and $S_k^r$ are, respectively, the $r$-uniform matching and star with
$k$ edges.
  The general bounds are improved for $3$-uniform hypergraphs. We prove that
$\chi(M_k^3,2)=2k$, extending a special case of Alon-Frankl-Lov\'asz' theorem.
We also prove that $\chi(S_2^3,t)\le t+1$, which is sharp for $t=2,3$. This is
a corollary of a more general result. We define $H^{[1]}$ as the 1-intersection
graph of $H$, whose vertices represent hyperedges and whose edges represent
intersections of hyperedges in exactly one vertex. We prove that $\chi(H)\le
\chi(H^{[1]})$ for any $3$-uniform hypergraph $H$ (assuming $\chi(H^{[1]})\ge
2$). The proof uses the list coloring version of Brooks' theorem.
