Suppose \((G,\Gamma)\) is a locally finite graph (i.e. \(G\) is a non-empty set and \(\Gamma\) is a function from \(G\) to \(\cal P(G)\) such that for each \(x\in G\), \(\Gamma(x)\) and \(\Gamma^{-1}\{x\}\) are finite), \(K\) is a finite set of integers, and \(T\) is a function mapping subsets of \(K\) into subsets of \(K\). If for each finite subgraph \((A,\Gamma_A)\) there is a function \(\psi\) such that for each \(x\in A\), \(\psi(x)\in T(\psi[\Gamma_A(x)])\), then there is a function \(\phi\) such that for all \(x\in G\), \(\phi(x)\in T(\phi[\Gamma(x)])\).

\(C(\infty,< \aleph_{0})\):  Every set of non-empty finite  sets  has  a choice function.

If  \(n\in\omega\), \(n\ge 2\) and \(N \subseteq \{ 1, 2, \ldots , n-1 \}\), \(N \neq\emptyset\), \(MC(\infty,n, N)\):  If \(X\) is any set of \(n\)-element sets then  there is  a function \(f\) with domain \(X\) such that for all \(A\in X\), \(f(A)\subseteq A\) and \(|f(A)|\in N\).