Poset Saturation of Unions of Chains

A family G G of sets is a(n induced) copy of a poset P=(P, ) P = ( P , ⩽ ) if there exists a bijection b:P→ G b : P → G such that p q p ⩽ q holds if and only if b(p)⊂ b(q) b ( p ) ⊂ b ( q ) . The induced saturation number sat^*(n,P) sat ∗ ( n , P ) is the minimum size of a family F⊆ 2^[n] F ⊆ 2 [ n ] that does not contain any copy of P P , but for any G∈ 2^[n] F G ∈ 2 [ n ] \ <