Gluing posets and the dichotomy of poset saturation numbers

Abstract Given a finite poset , we say that a family of subsets of is ‐saturated if does not contain an induced copy of , but adding any other set to creates an induced copy of . The saturation number of is the size of the smallest ‐saturated family with ground set . The saturation number for posets is known to exhibit a dichotomy: it is either bounded or it has a growth rate of at least . Determining which posets have bounded saturation number is a major open problem. In this paper we consider a ‘gluing’ operation, formed from two finite posets and by setting all elements of to be below all elements of . We show that (under some mild assumptions) this operation preserves bounded and unbounded saturation number. This is the first such ‘new from old’ poset construction to be found. As an application, we show that for any poset one may add at most 3 elements to to obtain a poset whose saturation number growth is at most linear: this may be viewed as a step towards the other major open problem in the area, namely the conjecture that every finite poset has growth that is at most linear. We also consider the poset equivalent of weak saturation for graphs: for each finite poset , we determine exactly the minimum size of a percolating family for .