Umeå universitets logga

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 (Formula presented). In this paper we improve this general result showing that either (Formula presented). 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 (Formula presented) we have (Formula presented); so if true this conjecture implies our result is tight up to a multi-plicative 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 (Formula presented). We prove that this latter conjecture is true for a certain class of posets P.