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.

Let \(E\) be a set and \(f: E\to E\), then \(f\) has a fixed point if and only if \(E\) is not the union of three mutually disjoint sets \(E_1\), \(E_2\) and \(E_3\) such that \(E_i \cap f(E_i) = \emptyset\) for \(i=1, 2, 3\).