  We consider the asymptotic minimum density $f(s,k)$ of monotone $k$-subwords
of words over a totally ordered alphabet of size $s$. The unrestricted alphabet
case, $f(\infty,k)$, is well-studied, known for $f(\infty,3)$ and
$f(\infty,4)$, and, in particular, conjectured to be rational for all $k$. Here
we determine $f(2,k)$ for all $k$ and determine $f(3,3)$, which is already
irrational. We describe an explicit construction for all $s$ which is
conjectured to yield $f(s,3)$. Using our construction and flag algebra, we
determine $f(4,3),f(5,3),f(6,3)$ up to $10^{-3}$ yet argue that flag algebra,
regardless of computational power, cannot determine $f(5,3)$ precisely.
Finally, we prove that for every fixed $k \ge 3$, the gap between $f(s,k)$ and
$f(\infty,k)$ is $\Theta(\frac{1}{s})$.
