SATURATION FOR THE BUTTERFLY POSET

Abstract

Given a finite poset P, we call a family F of subsets of [n] 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 induced saturated number of P, denoted by sat∗(n,P), is the size of the smallest P‐saturated family with ground set [n]. In this paper, we are mainly interested in the four‐point poset called the butterfly. Ferrara et al. showed that the saturation number for the butterfly lies between log2n and n2. We give a linear lower bound of n+1. We also prove some other results about the butterfly and the poset N.