The induced saturation problem for posets

For a fixed posetP, 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 S̸∈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) ⩾ log₂n. 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}. Ourproof 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.