Induced Saturation for Complete Bipartite Posets

Given s, t, a complete bipartite poset Kₒ, ₓ 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 F2^n is called induced Kₒ, ₓ-saturated if (F, ) contains no induced copy of Kₒ, ₓ, whereas adding any set from 2^n to F creates an induced Kₒ, ₓ. Let sat^* (n, Kₒ, ₓ) denote the smallest size of an induced Kₒ, ₓ-saturated family F2^n. We show that sat^* (n, Kₒ, ₓ) =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 Kₒ, ₂ with s.