  We study the separability problem for automatic relations (i.e., relations on
finite words definable by synchronous automata) in terms of recognizable
relations (i.e., finite unions of products of regular languages). This problem
takes as input two automatic relations $R$ and $R'$, and asks if there exists a
recognizable relation $S$ that contains $R$ and does not intersect $R'$. We
show this problem to be undecidable when the number of products allowed in the
recognizable relation is fixed. In particular, checking if there exists a
recognizable relation $S$ with at most $k$ products of regular languages that
separates $R$ from $R'$ is undecidable, for each fixed $k \geq 2$. Our proofs
reveal tight connections, of independent interest, between the separability
problem and the finite coloring problem for automatic graphs, where colors are
regular languages.
