Induced saturation for complete bipartite posets

Given s, t E N, 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 c 2[n] is called induced Ks,t-saturated if (F, c) contains no induced copy of Ks,t, whereas adding any set from 2[n]\F to F creates an induced Ks,t. Let sat & lowast;(n, Ks,t) denote the smallest size of an induced Ks,t-saturated family F c 2[n]. It was conjectured that sat & lowast;(n, Ks,t) is superlinear in n for certain values of sand t. In this paper, we show that sat & lowast;(n, Ks,t) = O(n) for all fixed s, t E N. Moreover, we prove a linear lower bound on sat & lowast;(n, P) for a large class of posets P, particularly for Ks,2 with s E N. (c) 2025 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).