Induced Poset Saturation

Given a fixed poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-free if it does not contain an (induced) copy of $\mathcal P$. And we say that $F$ is $\mathcal P$-saturated if it is maximal $\mathcal P$-free. How small can a $\mathcal P$-saturated family be? The smallest such size is the induced saturation number of $\mathcal P$, $\text{sat}^*(n, \mathcal P)$. Even for very small posets, the question of the growth speed of $\text{sat}^*(n,\mathcal P)$ seems to be hard. We present background on this problem and some recent results.