  The induced Ramsey number $r_{\mathrm{ind}}(F)$ of a $k$-uniform hypergraph
$F$ is the smallest natural number $n$ for which there exists a $k$-uniform
hypergraph $G$ on $n$ vertices such that every two-coloring of the edges of $G$
contains an induced monochromatic copy of $F$. We study this function, showing
that $r_{\mathrm{ind}}(F)$ is bounded above by a reasonable power of $r(F)$. In
particular, our result implies that $r_{\mathrm{ind}}(F) \leq 2^{2^{ct}}$ for
any $3$-uniform hypergraph $F$ with $t$ vertices, mirroring the best known
bound for the usual Ramsey number. The proof relies on an application of the
hypergraph container method.
