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) ₂ n\). In this paper we improve this general result showing that either \ (sat^* (n, P) =O (1) \) or \ (sat^* (n, P) \ 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, ) = (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