The saturation number of induced subposets of the Boolean lattice

Abstract

Given a poset P, a family F of elements in the Boolean lattice is said to be P-saturated if (1) F contains no copy of P as a subposet and (2) every proper superset of F contains a copy of P as a subposet. The maximum size of a P-saturated family is denoted by La(n, P), which has been studied for a number of choices of P. The minimum size of a P-saturated family, sat(n, P), was introduced by Gerbner et al. (2013), and parallels the deep literature on the saturation function for graphs.We introduce and study the concept of saturation for induced subposets. As opposed to induced saturation in graphs, the above definition of saturation for posets extends naturally to the induced setting. We give several exact results and a number of bounds on the induced saturation number for several small posets. We also use a transformation to the biclique cover problem to prove a logarithmic lower bound for a rich infinite family of target posets. †

Problems in extremal graphs and poset theory

“…In this chapter, we provide a lower bound result regarding induced-A k+1 -saturated families, which improves the result given in Ferrara et al [19]. Recall that sat * (n, A k+1 ) is the size of the smallest induced-A k+1 -saturated family in the n-dimensional Boolean lattice, B n , and Theo-…”

“…Recall Theorem 1.3.6, where bounds for sat * (n, P) were given for posets P = V 2 , D 2 , O 4 . Ferrara et al [19] also proved that for the poset N , we have log 2 n ≤ sat * (n, N ) ≤ 2n. In their study of induced-P-saturation, the authors conjectured the following:…”

“…Ferrara et al proved the following result regarding induced-A k+1 -saturation. Theorem 1.3.6 (Ferrara et al [19]). If n > k > 1, then…”

“…Ferrara et al [19] introduced and studied the concept of sat * (n, P) for various posets P. Our interest lies in studying induced saturation of the k-antichain A k . Let A k denote the k-antichain, which is the poset of k elements in which no pair of elements is comparable.…”

“…introduced the concept of studying the minimum size of a family of subsets of the n-set avoiding an induced poset, called induced-poset-saturation [19]. In particular, the authors provided a lower bound for the size of an induced-antichain poset and we improve on their lower bound result.…”