Cutsets in $\mathcal{P}(X)$

For any set $X$, $\mathcal{P}(X)$ denotes the collection of all subsets of $X$, ordered by inclusion. A {\it cutset} in $\mathcal{P}(X)$ is a subset of $\mathcal{P}(X)$ which meets every maximal chain of $\mathcal{P}(X)$. A cutset is non-trivial if it does not contain $X$ or the empty set. Our main result is the following. Theorem 1: Let $X$ be an infinite set of cardinality $\kappa$. Every non-trivial cutset in $\mathcal{P}(X)$ contains a chain of cardinality ${\kappa }^{+}$ and an antichain of cardinality $2^{\kappa }$.