Axiom Of Choice

We have the following indirect implication of form equivalence classes:

325 \(\Rightarrow\) 124 given by the following sequence of implications, with a reference to its direct proof:

Implication	Reference
325 \(\Rightarrow\) 17	clear
17 \(\Rightarrow\) 124	Hilbertraume mit amorphen Basen (English summary), Brunner, N. 1984a, Compositio Math.

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

Howard-Rubin Number	Statement
325:	Ramsey's Theorem II: \(\forall n,m\in\omega\), if A is an infinite set and the family of all \(m\) element subsets of \(A\) is partitioned into \(n\) sets \(S_{j}, 1\le j\le n\), then there is an infinite subset \(B\subseteq A\) such that all \(m\) element subsets of \(B\) belong to the same \(S_{j}\). (Also, see Form 17.)
17:	Ramsey's Theorem I: If \(A\) is an infinite set and the family of all 2 element subsets of \(A\) is partitioned into 2 sets \(X\) and \(Y\), then there is an infinite subset \(B\subseteq A\) such that all 2 element subsets of \(B\) belong to \(X\) or all 2 element subsets of \(B\) belong to \(Y\). (Also, see Form 325.), Jech [1973b], p 164 prob 11.20.
124:	Every operator on a Hilbert space with an amorphous base is the direct sum of a finite matrix and a scalar operator. (A set is amorphous if it is not the union of two disjoint infinite sets.)

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