The induced saturation problem for posets

For a fixed poset $P$, a family $\mathcal{F}$ of subsets of $[n]$ is induced $P$-saturated if $\mathcal{F}$ does not contain an induced copy of $P$, but for every subset $S$ of $[n]$ such that $S\in ̸\mathcal{F}$, $P$ is an induced subposet of $\mathcal{F}\cup \{S\}$. The size of the smallest such family $\mathcal{F}$ is denoted by ${\text{sat}}^{\ast }(n,P)$. Keszegh, Lemons, Martin, Pálvölgyi and Patkós [Journal of Combinatorial Theory Series A, 2021] proved that there is a dichotomy of behaviour for this parameter: given any poset $P$, either ${\text{sat}}^{\ast }(n,P)=O(1)$ or ${\text{sat}}^{\ast }(n,P)\ge {\log }_{2}n$. In this paper we improve this general result showing that either ${\text{sat}}^{\ast }(n,P)=O(1)$ or ${\text{sat}}^{\ast }(n,P)\ge \min \{2\sqrt{n},n\mathrm{/}2+1\}$. Our proof makes use of a Turán-type result for digraphs. Curiously, it remains open as to whether our result is essentially best possible or not. On the one hand, a conjecture of Ivan states that for the so-called diamond poset $\mathrm{◊}$ we have ${\text{sat}}^{\ast }(n,\mathrm{◊})=\mathrm{\Theta }(\sqrt{n})$; so if true this conjecture implies our result is tight up to a multiplicative constant. On the other hand, a conjecture of Keszegh, Lemons, Martin, Pálvölgyi and Patkós states that given any poset $P$, either ${\text{sat}}^{\ast }(n,P)=O(1)$ or ${\text{sat}}^{\ast }(n,P)\ge n+1$. We prove that this latter conjecture is true for a certain class of posets $P$.