Induced saturation for complete bipartite posets

Given $s,t\in N$, a complete bipartite poset $K_{s,t}$ 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 $F\subseteq 2^{\left[n\right]}$ is called induced $K_{s,t}$-saturated if $(F,\subseteq )$ contains no induced copy of $K_{s,t}$, whereas adding any set from $2^{\left[n\right]}﹨F$ to $F$ creates an induced $K_{s,t}$. Let ${\mathrm{sat}}^{⁎}(n,K_{s,t})$ denote the smallest size of an induced $K_{s,t}$-saturated family $F\subseteq 2^{\left[n\right]}$. It was conjectured that ${\mathrm{sat}}^{⁎}(n,K_{s,t})$ is superlinear in n for certain values of s and t. In this paper, we show that ${\mathrm{sat}}^{⁎}(n,K_{s,t})=O\left(n\right)$ for all fixed $s,t\in N$. Moreover, we prove a linear lower bound on ${\mathrm{sat}}^{⁎}(n,P)$ for a large class of posets $P$, particularly for $K_{s,2}$ with $s\in N$.