Linear Saturation for \(\mathcal{N}\) via Butterflies

Abstract. Given a finite poset [Formula: see text], how small can a family [Formula: see text] of subsets of [Formula: see text] be such that [Formula: see text] does not contain an induced copy of [Formula: see text], but [Formula: see text] contains such a copy for all [Formula: see text]? This is known as the induced saturation number of [Formula: see text], denoted by [Formula: see text]. The main conjecture in this area is that the induced saturation number for any poset is either bounded or linear. In this paper we establish linearity for the induced saturation number of the 4-point poset [Formula: see text]. Previously, it was known that [Formula: see text]. We show that [Formula: see text]. A crucial role in the proof is played by a structural feature of [Formula: see text]-saturated families, namely that if the family contains two antichains, one completely above the other, then it must also contain a “middle” point that is greater than one antichain and less than the other.