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.

\((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element  sets has a choice function.

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\), \(A\) has an infinite subset \(B\) such that for every \(n\in\omega\), \(n>0\), the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco [1998b]

A Form of Restricted Choice for Families of Finite Sets: For every infinite set \(A\) and every \(n\in\omega\), there is an infinite subset \(B\) of \(A\) such the set of all \(n\) element subsets of \(B\) has a choice function. De la Cruz/Di Prisco} [1998b]

(For \(n\in\omega\), \(n\ge 2\).)  For every infinite set \(X\), there is an infinite \(Y \subseteq X\) such that the set of all \(n\)-element subsets of \(Y\) has a choice function.