C O ] 5 J ul 2 02 1 Induced and non-induced poset saturation problems

A subfamily G ⊆ F ⊆ 2[n] 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 ≤P q implies i(p) ⊆ i(q). In the case where in addition p ≤P q 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 ⊆ 2[n] 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 ∈ 2[n] \ 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.