Saturation for Sums of Posets and Antichains

Given a finite poset P, we say that a family F of subsets of n is P-saturated if F does not contain an induced copy of P, but adding any other set to F creates an induced copy of P. The saturation number of P is the size of the smallest P-saturated family with ground set n. The saturation numbers have been shown to exhibit a dichotomy: for any poset, the saturation number is either bounded, or at least 2 n. The general conjecture is that in fact, the saturation number for any poset is either bounded, or at least linear. The linear sum of two posets P₁ and P₂, dented by P₁* P₂, is defined as the poset obtained from a copy of P₁ placed completely on top of a copy of P₂. In this paper we show that the saturation number of P₁* Aₖ* P₂ is always at least linear, for any P₁, P₂ and k2, where Aₖ is the antichain of size k. This is a generalisation of the recent result that the saturation number for the diamond is linear (in that case P₁ and P₂ are both the single point poset, and k=2). We also show that, with the exception of chains which are known to have bounded saturation number, the saturation number for all complete multipartite posets is linear.