The induced saturation problem for posets

For a fixed poset \(P\), a family \( F\) of subsets of \([n]\) is induced \(P\)-saturated if \( F\) does not contain an induced copy of \(P\), but for every subset \(S\) of \([n]\) such that \( S ∈ F\), \(P\) is an induced subposet of \( F ∪ \{S\}\). The size of the smallest such family \( F\) is denoted by \(sat^* (n,P)\). Keszegh, Lemons, Martin, Pálvölgyi and Patkós [Journal of Combinatorial Theory Series A, 2021] proved that there is a dichotomy of behaviour for this parameter: given any poset \(P\), either \(sat^* (n,P)=O(1)\) or \(sat^* (n,P)≥ log _2 n\). In this paper we improve this general result showing that either \(sat^* (n,P)=O(1)\) or \(sat^* (n,P) ≥ min\{ 2 √n, n/2+1\}\). Our proof makes use of a Turán-type result for digraphs.Curiously, it remains open as to whether our result is essentially best possible or not. On the one hand, a conjecture of Ivan states that for the so-called diamond poset \(\) we have \(sat^* (n,)=Θ (√n)\); so if true this conjecture implies our result is tight up to a multiplicative constant. On the other hand, a conjecture of Keszegh, Lemons, Martin, Pálvölgyi and Patkós states that given any poset \(P\), either \(sat^* (n,P)=O(1)\) or \(sat^* (n,P)≥ n+1\). We prove that this latter conjecture is true for a certain class of posets \(P\).Mathematics Subject Classifications: 06A07, 05D05Keywords: Partially ordered sets, saturation, Turán-type problems