Induced saturation for complete bipartite posets

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