  We study the phase transition from dense baryonic matter to dense quark
matter within the large-$N_{c}$ limit. By using simple constant speed of sound
equations of state for the two phases, we derive the scaling with $N_{c}$ of
the critical quark chemical potential $\mu_{c}^{\mathrm{crit}}$ for this phase
transition. While quark matter is strongly suppressed at large $N_{c}$, the
phase transition at a large but finite density could nevertheless be important
to determine the maximum mass of compact stars. In particular, in the range
$3\leq N_{c}\lesssim5.5$ the quark phase would take place in compact stars and
would lead to the formation of an unstable branch of hybrid stars. As a
consequence, the maximum mass is restricted to the range
$2.1M_{\odot}<M_{max}<3M_{\odot}$. For larger value of $N_{c}$, the phase
transition would occur at densities too high to be reached in the core of
compact stars. However, the very requirement that it occurs (although at a very
large density) translates into interesting constraints on the stiffness of the
baryonic phase: its speed of sound must exceed $\sqrt{1/3}$.
