\(PKW(\infty,\infty,\infty)\): For every infinite family \(X\) of sets each of which has at least two elements, there is an infinite subfamily \(Y\) of \(X\) and a function \(f\) such that for all \(y\in Y\), \(f(y)\) is a non-empty proper subset of \(y\).

\(PKW(\aleph_{0},\ge 2,\infty)\), Partial Kinna-Wagner Principle:  For every denumerable family \(F\) such that for all \(x\in F\), \(|x|\ge 2\), there is an infinite subset \(H\subseteq F\) and a function \(f\) such that for all \(x\in H\), \(\emptyset\neq f(x) \subsetneq x\).

\(PUT(\aleph_{0},2,\aleph_{0})\):  The union of a denumerable family of pairwise disjoint pairs has a denumerable subset.