On the Minimum Density of Monotone Subwords

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(∞, k), is well-studied, known for f(∞, 3) and f(∞, 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 ≥ 3, the gap between f(s, k) and f(∞, k) is [Formula Presented]^..