Every field \(F\) and every vector space \(V\) over \(F\) has the property that each linearly independent set \(A\subseteq V\) can be extended to a basis. H.Rubin/J.~Rubin [1985], pp 119ff.

\((\forall n\in\omega - \{0\}) MC(\infty,\infty \), relatively prime to \(n\)): \(\forall n\in\omega -\{0\}\), if \(X\) is a set of non-empty sets, then  there  is  a function \(f\) such that for all \(x\in X\), \(f(x)\) is a non-empty, finite subset of \(x\) and \(|f(x)|\) is relatively prime to \(n\).

\(MC(\infty,\infty,\mathrm{odd})\): For every set \(X\) of  sets such that for all \(x\in X\), \(|x|\ge 1\), there is a function \(f\) such that  for every \(x\in X\), \(f(x)\) is a finite, non-empty subset of \(x\) and \(|f(x)|\) is odd.