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.

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

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

\(KW(WO,<\aleph_0)\),  The Kinna-Wagner Selection Principle for a well ordered family of finite sets: For every well ordered set \(M\) of finite sets there is a function \(f\) such that for all \(A\in M\), if \(|A|>1\)  then \(\emptyset\neq f(A)\subsetneq A\). (See Form 15.)