  We investigate the maximal size of an increasing subset among points randomly
sampled from certain probability densities. Kerov and Vershik's celebrated
result states that the largest increasing subset among $N$ uniformly random
points on $[0,1]^2$ has size asymptotically $2\sqrt{N}$. More generally, the
order $\Theta(\sqrt{N})$ still holds if the sampling density is continuous. In
this paper we exhibit two sufficient conditions on the density to obtain a
growth rate equivalent to any given power of $N$ greater than $\sqrt{N}$, up to
logarithmic factors. Our proofs use methods of slicing the unit square into
appropriate grids, and investigating sampled points appearing in each box.
