Implications

We have the following indirect implication of form equivalence classes:

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

Implication Reference
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
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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