Induced and non-induced poset saturation problems

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