  Shift spaces with the specification property are intrinsically ergodic, i.e.
they have a unique measure of maximal entropy. This can fail for shifts with
the weaker almost specification property. We define a new property called
one-sided almost specification, which lies in between specification and almost
specification, and prove that it guarantees intrinsic ergodicity if the
corresponding mistake function g is bounded. We also show that uniqueness may
fail for unbounded g such as log log n. Our results have consequences for
almost specification: we prove that almost specification with g=1 implies
one-sided almost specification (with g=1), and hence uniqueness. On the other
hand, the second author showed recently that almost specification with g=4 does
not imply uniqueness. This leaves open the question of whether almost
specification implies intrinsic ergodicity when g=2 or g=3.
