The saturation number for the diamond is linear

Abstract For a fixed poset we say that a family is ‐saturated if it does not contain an induced copy of , but whenever we add a new set to , we form an induced copy of . The size of the smallest such family is denoted by . For the diamond poset (the two‐dimensional Boolean lattice), while it is easy to see that the saturation number is at most , the best known lower bound has stayed at since the introduction of the area of poset saturation. In this paper, we prove that , establishing that the saturation number for the diamond is linear. The proof uses a result about certain pairs of set systems.