  We prove that every non-minimal transitive subshift $X$ satisfying a mild
aperiodicity condition satisfies $\limsup c_n(X) - 1.5n = \infty$, and give a
class of examples which shows that the threshold of $1.5n$ cannot be increased.
As a corollary, we show that any transitive $X$ satisfying $\limsup c_n(X) - n
= \infty$ and $\limsup c_n(X) - 1.5n < \infty$ must be minimal. We also prove
some restrictions on the structure of transitive non-minimal $X$ satisfying
$\liminf c_n(X) - 2n = -\infty$, which imply unique ergodicity (for a periodic
measure) as a corollary, which extends a result of Boshernitzan from the
minimal case to the more general transitive case.
