Induced and non-induced poset saturation problems

A subfamily G⊆F⊆2n of sets is a non-induced (weak) copy of a poset P in F if there exists a bijection i:P→G such that p≤Pq implies i(p)⊆i(q). In the case where in addition p≤Pq holds if and only if i(p)⊆i(q), then G is an induced (strong) copy of P in F. We consider the minimum number sat(n,P) resp. sat⁎(n,P) of sets that a family F⊆2n can have without containing a non-induced induced copy of P and being maximal with respect to this property, i.e., the addition of any G∈2n∖F creates a non-induced induced copy of P. We prove for any finite poset P that sat(n,P)≤2|P|−2, a bound independent of the size n of the ground set. For induced copies of P, there is a dichotomy: for any poset P either sat⁎(n,P)≤KP for some constant depending only on P or sat⁎(n,P)≥log2⁡n. We classify several posets according to this dichotomy, and also show better upper and lower bounds on sat(n,P) and sat⁎(n,P) for specific classes of posets. Our main new tool is a special ordering of the sets based on the colexicographic order. It turns out that if P is given, processing the sets in this order and adding the sets greedily into our family whenever this does not ruin non-induced induced P-freeness, we tend to get a small size non-induced induced P-saturating family.