Induced Saturation for Complete Bipartite Posets

Given s,t, a complete bipartite poset Ks,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 Ks,t, whereas adding any set from 2[n] to F creates an induced Ks,t. Let sat*(n,Ks,t) denote the smallest size of an induced Ks,t-saturated family F⊆2[n]. We show that sat*(n,Ks,t)=O(n) for all s,t. Moreover, we prove a linear lower bound on sat*(n,P) for a large class of posets P, particularly for Ks,2 with s.