Improved bounds for induced poset saturation

Abstract

Given a finite poset P, a family F of elements in the Boolean lattice is induced-P-saturated if F contains no copy of P as an induced subposet but every proper superset of F contains a copy of P as an induced subposet. The minimum size of an induced-P-saturated family in the n-dimensional Boolean lattice, denoted sat * (n, P), was first studied by Ferrara et al. (2017).Our work focuses on strengthening lower bounds. For the 4point poset known as the diamond, we prove sat * (n, D 2 ) ≥ √ n, improving upon a logarithmic lower bound. For the antichain with k + 1 elements, we prove sat * (n, A k+1 ) ≥ (1 − o k (1)) kn log 2 k , improving upon a lower bound of 3n − 1 for k ≥ 3.

Saturation for the Butterfly Poset

“…This approach was also used by Martin, Smith and Walker [4] in their analysis of the saturation number of another poset, the diamond. Proof.…”

Saturation problems in the Ramsey theory of graphs, posets and point sets

“…This result was extended to the case of families without 𝑘-matchings by Bucić et al [2]. In the setting of forbidden (induced or non-induced) posets in the Boolean lattice the saturation problem has been investigated by Ferrara et al [9], and further results in this direction were obtained in [17] and [12].…”

Induced and non-induced poset saturation problems

“…This upper bound was generalized to all k by Morrison, Noel, and Scott [12], proving C • 2 (1−δ)k where δ = 1 − log 2 15 4 ≈ 0.02. Later, the induced version sat * (n, P ) was studied by Ferrara et al [3] and by Martin, Smith, and Walker [10]. In [3], it was shown for a number of other posets P that sat(n, P ) is bounded by some constant independent of n, while sat * (n, P ) was shown to be unbounded for all these posets.…”