Saturation for Sums of Posets and Antichains

Given a finite poset $\mathcal P$, we say that a family $\mathcal F$ of subsets of $[n]$ is $\mathcal P$-saturated if $\mathcal F$ does not contain an induced copy of $\mathcal P$, but adding any other set to $\mathcal F$ creates an induced copy of $\mathcal P$. The saturation number of $\mathcal P$ is the size of the smallest $\mathcal P$-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 $2\sqrt n$. 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 $\mathcal P_1$ and $\mathcal P_2$, dented by $\mathcal P_1*\mathcal P_2$, is defined as the poset obtained from a copy of $\mathcal P_1$ placed completely on top of a copy of $\mathcal P_2$. In this paper we show that the saturation number of $\mathcal P_1*\mathcal A_k*\mathcal P_2$ is always at least linear, for any $\mathcal P_1$, $\mathcal P_2$ and $k\geq2$, where $\mathcal A_k$ 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 $\mathcal P_1$ and $\mathcal P_2$ 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.