Implications

We have the following indirect implication of form equivalence classes:

45-n \(\Rightarrow\) 336-n
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
45-n \(\Rightarrow\) 336-n clear

Here are the links and statements of the form equivalence classes referenced above:

Howard-Rubin Number Statement
45-n:

If \(n\in\omega-\{0,1\}\), \(C(\infty,n)\): Every set of \(n\)-element sets has a choice function.

336-n:

(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.

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