Implications

We have the following indirect implication of form equivalence classes:

336-n \(\Rightarrow\) 390
given by the following sequence of implications, with a reference to its direct proof:

Implication Reference
336-n \(\Rightarrow\) 64 Weak choice principles, De la Cruz, O. 1998a, Proc. Amer. Math. Soc.
64 \(\Rightarrow\) 390 clear

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

Howard-Rubin Number Statement
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.

64:

\(E(I,Ia)\) There are no amorphous sets. (Equivalently, every infinite set is the union of two disjoint infinite sets.)

390:

Every infinite set can be partitioned either into two infinite sets or infinitely many sets, each of which has at least two elements. Ash [1983].

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