Given a set X, the power set \(\mathcal {P}(X)\) , and a finite poset \((P,\le _P)\) , a family \(\mathcal {F}\subseteq \mathcal {P}(X)\) is said to be induced-P-free if there is no injection \(\varphi : P\rightarrow \mathcal {F}\) such that \(\varphi (p)\subseteq \varphi (q)\) if and only if \(p\le _{P} q\) for every \(p,q \in P\) . The family \(\mathcal {F}\) 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 \(\mathcal {P}(X)\) is denoted \(\textrm{sat}^*(n,P)\) . The poset \(2C_2\) is two incomparable 2-chains (the Hasse diagram is two vertex-disjoint edges) and Keszegh, Lemons, Martin, Pálvölgyi, and Patkós proved that \(n+2\le \textrm{sat}^*(n,2C_2)\le 2n\) 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 \(\mathcal {P}(X)\) where \(|X|=n\) .
