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

TL;DR

This paper investigates the structure of cutsets in the power set lattice of an infinite set, revealing that non-trivial cutsets contain large chains and antichains, highlighting their complex combinatorial properties.

Contribution

It establishes that every non-trivial cutset in the power set lattice of an infinite set contains large chains and antichains, providing new insights into their structure.

Findings

Non-trivial cutsets contain a chain of size 1^+

Non-trivial cutsets contain an antichain of size 2^1

Results apply to infinite sets of arbitrary cardinality

Abstract

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 $2^{\kappa}$.