Minimal Diamond-Saturated Families

For a given fixed poset $\mathcal P$ we say that family of subsets $[n]$ is P$-saturated if it does not contain an induced copy P$, but whenever add to new set, formed. The size the smallest such denoted by $\text{sat}^*(n, \mathcal P)$. diamond D_2$ (the two-dimensional Boolean lattice), Martin, Smith and Walker proved $\sqrt n\leq\text{sat}^*(n, D_2)\leq n+1$. In this paper prove D_2)\geq (2\sqrt2-o(1))\sqrt n$. We also explore properties diamond-saturated $c\sqrt n$, for constant $c$, would have have.