  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\not \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 $\text{sat}^* (n,P)$. Keszegh, Lemons, Martin,
P\'alv\"olgyi and Patk\'os [Journal of Combinatorial Theory Series A, 2021]
proved that there is a dichotomy of behaviour for this parameter: given any
poset $P$, either $\text{sat}^* (n,P)=O(1)$ or $\text{sat}^* (n,P)\geq \log _2
n$. In this paper we improve this general result showing that either
$\text{sat}^* (n,P)=O(1)$ or $\text{sat}^* (n,P) \geq \min\{ 2 \sqrt{n},
n/2+1\}$. Our proof makes use of a Tur\'an-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 $\Diamond$ we have $\text{sat}^* (n,\Diamond)=\Theta
(\sqrt{n})$; 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, P\'alv\"olgyi and Patk\'os states that given any poset $P$, either
$\text{sat}^* (n,P)=O(1)$ or $\text{sat}^* (n,P)\geq n+1$. We prove that this
latter conjecture is true for a certain class of posets $P$.

Combinatorics