Minimal Diamond-Saturated Families
Maria-Romina Ivan
TL;DR
The paper studies the induced saturation number $\text{sat}^*(n, \mathcal{D}_2)$ for the diamond poset in the Boolean lattice, improving the best-known lower bound from $\sqrt{n}$ to $$(4-o(1))\sqrt{n}$$ via a global analysis that leverages minimal and maximal elements and a bipartite-graph framework built from diamond configurations. It expands the understanding of the structure of potential size-$c\sqrt{n}$ diamond-saturated families, suggesting possible Dilworth-type decompositions and complement-invariance patterns, and it discusses whether $\text{sat}^*(n, \mathcal{D}_2)$ could be as small as $O(\sqrt n)$, ultimately showing that certain extreme configurations are impossible. An update resolves a key structural Question 5 in the negative, reinforcing the view that the true growth rate is likely $\Theta(\sqrt n)$ and guiding future construction efforts for tight saturating families. Overall, the results advance the extremal poset saturation landscape by tightening constants and clarifying the global structural features such saturated families would need to exhibit.
Abstract
For a given fixed poset $\mathcal P$ we say that a family of subsets of $[n]$ is $\mathcal P$-saturated if it does not contain an induced copy of $\mathcal P$, but whenever we add to it a new set, an induced copy of $\mathcal P$ is formed. The size of the smallest such family is denoted by $\text{sat}^*(n, \mathcal P)$. For the diamond poset $\mathcal D_2$ (the two-dimensional Boolean lattice), Martin, Smith and Walker proved that $\sqrt n\leq\text{sat}^*(n, \mathcal D_2)\leq n+1$. In this paper we prove that $\text{sat}^*(n, \mathcal D_2)\geq (4-o(1))\sqrt n$. We also explore the properties that a diamond-saturated family of size $c\sqrt n$, for a constant $c$, would have to have.