Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 109 / 1 / i

The hypothesis says exactly that $\mathcal{A}$ is a k-Sperner family with $k=2$: it contains no three members forming a strict inclusion chain. Every maximal chain in a Boolean lattice consequently contains at most two members of $\mathcal{A}$, so the same random-chain argument gives $\lambda (\mathcal{A})\le 2$. For a fixed amount of Lubell mass, cardinality is maximized by using the levels with the two largest binomial coefficients. The Erdős theorem on k-Sperner families therefore givesThe bound is attained by taking the union of two levels having those sizes.