  We give a lower bound on the growth of a subshift based on a simple condition
on the set of forbidden patterns defining that subshift. Aubrun et Al. showed a
similar result based on the Lov\'asz Local Lemma for subshift over any
countable group and Bernshteyn extended their approach to deduce, amongst other
things, some lower bound on the exponential growth of the subshift. However,
our result has a simpler proof, is easier to use for applications, and provides
better bounds on the applications from their articles (although it is not clear
that our result is stronger in general).
  In the particular case of subshift over $\mathbb{Z}$ a similar but weaker
condition given by Miller was known to imply nonemptiness of the associated
shift. Pavlov used the same approach to provide a condition that implied
exponential growth. We provide a version of our result for this particular
setting and it is provably strictly stronger than the result of Pavlov and the
result of Miller (and, in practice, leads to considerable improvement in the
applications).
  We also apply our two results to a few different problems including strongly
aperiodic subshifts, nonrepetitive subshifts, and Kolmogorov complexity of
subshifts.
