Given a finite poset $P$ we ask how small a family of subsets of $[n]$ can be such that it does not contain an induced copy of the poset, but adding any other subset creates such a copy. This number is called the saturation number of $P$, denoted by $\operatorname{sat}^*(n,P)$. Despite the apparent similarity to the saturation for graphs, this notion is vastly different. For example, it has been shown that the saturation numbers exhibit a dichotomy: for any poset, the saturation number is either bounded, or at least $2 n^{1/2}$. In fact, it is believed that the saturation number is always bounded or exactly linear. In this talk we will be discussing the most recent advances in this field, with the focus on the diamond poset, whose saturation number was unknown until recently.
Joint with Sean Jaffe.