Improved bounds for induced poset saturation

Abstract

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 ${\mathrm{sat}}^{\ast }(n,\mathcal{P})$, was first studied by Ferrara et al. (2017). Our work focuses on strengthening lower bounds. For the 4-point poset known as the diamond, we prove ${\mathrm{sat}}^{\ast }(n,{\mathcal{D}}_2)\ge \sqrt{n}$, improving upon a logarithmic lower bound. For the antichain with $k+1$ elements, we prove ${\mathrm{sat}}^{\ast }(n,{\mathcal{A}}_{k+1})\ge (1-o_k(1))\frac{kn}{{\log }_{2}k}$, improving upon a lower bound of $3n-1$ for $k\ge 3$.