  If $\mathcal{A}$ is a finite set (alphabet), the shift dynamical system
consists of the space $\mathcal{A}^{\mathbb{N}}$ of sequences with entries in
$\mathcal{A}$, along with the left shift operator $S$. Closed $S$-invariant
subsets are called subshifts and arise naturally as encodings of other systems.
In this paper, we study the number of ergodic measures for transitive subshifts
under a condition ("regular bispecial condition") on the possible extensions of
words in the associated language. Our main result shows that under this
condition, the subshift can support at most $\frac{K+1}{2}$ ergodic measures,
where $K$ is the limiting value of $p(n+1)-p(n)$, and $p$ is the complexity
function of the language. As a consequence, we answer a question of
Boshernitzan from `84, providing a combinatorial proof for the bound on the
number of ergodic measures for interval exchange transformations.
