  We answer a question posed by Hirschfeldt and Jockusch by showing that
whenever $k > \ell$, Ramsey's theorem for singletons and $k$-colorings,
$\mathsf{RT}^1_k$, is not strongly computably reducible to the stable Ramsey's
theorem for $\ell$-colorings, $\mathsf{SRT}^2_\ell$. Our proof actually
establishes the following considerably stronger fact: given $k > \ell$, there
is a coloring $c : \omega \to k$ such that for every stable coloring $d :
[\omega]^2 \to \ell$ (computable from $c$ or not), there is an infinite
homogeneous set $H$ for $d$ that computes no infinite homogeneous set for $c$.
This also answers a separate question of Dzhafarov, as it follows that the
cohesive principle, $\mathsf{COH}$, is not strongly computably reducible to the
stable Ramsey's theorem for all colorings, $\mathsf{SRT}^2_{<\infty}$. The
latter is the strongest partial result to date in the direction of giving a
negative answer to the longstanding open question of whether $\mathsf{COH}$ is
implied by the stable Ramsey's theorem in $\omega$-models of $\mathsf{RCA}_0$.
