# The Saturation Number of Induced Subposets of the Boolean Lattice

## TL;DR

This paper investigates the minimum size of induced P-saturated families within the Boolean lattice, providing exact results, bounds, and a transformation to biclique cover problems to establish lower bounds.

## Contribution

It introduces the concept of induced saturation for posets, extending existing saturation theory, and offers new bounds and exact results for various small posets.

## Key findings

- Exact results and bounds for induced saturation numbers of small posets
- A transformation to biclique cover problem for lower bounds
- Logarithmic lower bound for an infinite family of posets

## 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.