\(C(2^{\aleph_{0}},\subseteq{\Bbb R})\): If \(R\) is a relation on \({\Bbb R}\) such that for all \(x\in{\Bbb R}\), there is a \(y\in{\Bbb R}\) such that \(x\mathrel R y\), then there is a function \(f: {\Bbb R} \rightarrow{\Bbb R}\) such that for all \(x\in{\Bbb R}\), \(x\mathrel R f(x)\).

\(C(\aleph_{0},\infty,{\Bbb R})\): Every denumerable family of non-empty sets of reals  has a choice function. Jech [1973b], p 148 prob 10.1.

\(C(\aleph_0,\aleph_0,\Bbb R)\): Every denumerable set of non-empty denumerable subsets of \({\Bbb R}\) has a choice function.