Induced Saturation of the Poset 2C_2

TL;DR

This paper investigates the minimal size of families in the power set of a set that are maximally free of a specific poset structure called 2C2, improving known bounds and providing new constructions.

Contribution

It improves the lower bound on the size of induced-2C2-saturated families and offers numerous examples of families reaching the upper bound.

Findings

Lower bound on size improved to 3n/2 + 1/2

Constructed many families of size 2n that are induced-2C2-saturated

Confirmed the upper bound of 2n for certain families

Abstract

Given a set $X$, the power set $\mathbb{P}(X)$, and a finite poset $P$, a family $F\subset \mathbb{P}(X)$ is said to be induced-$P$-free if there is no injection $\phi: P\rightarrow \mathbb{F}$ such that $\phi(p)\subseteq\phi(q)$ if and only if $p\leq_{P} q$, for all $p, q \in P$. The family $F$ is induced-$P$-saturated if it is maximal with respect to being induced-$P$-free. If $n=|X|$, then the size of the smallest induced-$P$-saturated family in $\mathbb{P}(X)$ is denoted $sat(n,P)$. The poset $2C_2$ is two incomparable 2-chains (the Hasse diagram is two vertex-disjoint edges) and Keszegh, Lemons, Martin, P\'alv\"olgyi, and Patk\'os proved that $n+2\leq sat(n,2C_2)\leq 2n$ and gave one isomorphism class of an induced-$2C_2$-saturated family that achieves the upper bound. We show that the lower bound can be improved to $3n/2 + 1/2$ by examining the necessary structure of a saturated family. In addition, we provide many examples of induced-$2C_2$-saturated families of size $2n$ in $\mathbb{P}(X)$ where $|X|=n$.