  In combinatorics on words, a word w over an alphabet Sigma is said to avoid a
pattern p over an alphabet Delta if there is no factor x of w and no
non-erasing morphism h from Delta^* to Sigma^* such that h(p) = x. Bell and Goh
have recently applied an algebraic technique due to Golod to show that for a
certain wide class of patterns p there are exponentially many words of length n
over a 4-letter alphabet that avoid p. We consider some further consequences of
their work. In particular, we show that any pattern with k variables of length
at least 4^k is avoidable on the binary alphabet. This improves an earlier
bound due to Cassaigne and Roth.
