Improved bounds for induced poset saturation (Q2185221)

Summary: Given a finite poset \(\mathcal{P} \), a family \(\mathcal{F}\) of elements in the Boolean lattice is induced-\( \mathcal{P} \)-saturated if \(\mathcal{F}\) contains no copy of \(\mathcal{P}\) as an induced subposet but every proper superset of \(\mathcal{F}\) contains a copy of \(\mathcal{P}\) as an induced subposet. The minimum size of an induced-\( \mathcal{P} \)-saturated family in the \(n\)-dimensional Boolean lattice, denoted \(\text{sat}^*(n,\mathcal{P})\), was first studied by \textit{M. Ferrara} et al. [Discrete Math. 340, No. 10, 2479--2487 (2017; Zbl 1423.06006)]. Our work focuses on strengthening lower bounds. For the 4-point poset known as the diamond, we prove \(\text{sat}^*(n,\mathcal{D}_2)\geqslant\sqrt{n} \), improving upon a logarithmic lower bound. For the antichain with \(k+1\) elements, we prove \[\text{sat}^*(n,\mathcal{A}_{k+1})\geqslant \left(1-\frac{1}{\log_2k}\right)\frac{kn}{\log_2 k}\] for \(n\) sufficiently large, improving upon a lower bound of \(3n-1\) for \(k\geqslant 3\).