Minimal Diamond-Saturated Families

For a given fixed poset $\mathcal P$ we say that a family of subsets of $[n]$ is $\mathcal P$-saturated if it does not contain an induced copy of $\mathcal P$, but whenever we add to it a new set, an induced copy of $\mathcal P$ is formed. The size of the smallest such family is denoted by $\text{sat}^*(n, \mathcal P)$. For the diamond poset $\mathcal D_2$ (the two-dimensional Boolean lattice), Martin, Smith and Walker proved that $\sqrt n\leq\text{sat}^*(n, \mathcal D_2)\leq n+1$. In this paper we prove that $\text{sat}^*(n, \mathcal D_2)\geq (4-o(1))\sqrt n$. We also explore the properties that a diamond-saturated family of size $c\sqrt n$, for a constant $c$, would have to have.