  Given $s,t\in\mathbb{N}$, a complete bipartite poset $\mathcal{K}_{s,t}$ is a
poset, whose Hasse diagram consists of $s$ pairwise incomparable vertices in
the upper layer and $t$ pairwise incomparable vertices in the lower layer, such
that every vertex in the upper layer is larger than all vertices in the lower
layer. A family $\mathcal{F}\subseteq2^{[n]}$ is called induced
$\mathcal{K}_{s,t}$-saturated if $(\mathcal{F},\subseteq)$ contains no induced
copy of $\mathcal{K}_{s,t}$, whereas adding any set from
$2^{[n]}\backslash\mathcal{F}$ to $\mathcal{F}$ creates an induced
$\mathcal{K}_{s,t}$. Let $\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})$ denote the
smallest size of an induced $\mathcal{K}_{s,t}$-saturated family
$\mathcal{F}\subseteq2^{[n]}$. We show that
$\mathrm{sat}^{*}(n,\mathcal{K}_{s,t})=O(n)$ for all fixed $s,t\in\mathbb{N}$.
Moreover, we prove a linear lower bound on $\mathrm{sat}^{*}(n,\mathcal{P})$
for a large class of posets $\mathcal{P}$, particularly for $\mathcal{K}_{s,2}$
with $s\in\mathbb{N}$.
