We have the following indirect implication of form equivalence classes:
| Implication | Reference |
|---|---|
| 409 \(\Rightarrow\) 62 |
Short proof of a theorem of Rado on graphs, Foster, B. L. 1964, Proc. Amer. Math. Soc. note-152 |
| 62 \(\Rightarrow\) 61 | clear |
| 61 \(\Rightarrow\) 45-n | clear |
| 45-n \(\Rightarrow\) 33-n | clear |
| 33-n \(\Rightarrow\) 47-n | clear |
Here are the links and statements of the form equivalence classes referenced above:
| Howard-Rubin Number | Statement |
|---|---|
| 409: | Suppose \((G,\Gamma)\) is a locally finite graph (i.e. \(G\) is a non-empty set and \(\Gamma\) is a function from \(G\) to \(\cal P(G)\) such that for each \(x\in G\), \(\Gamma(x)\) and \(\Gamma^{-1}\{x\}\) are finite), \(K\) is a finite set of integers, and \(T\) is a function mapping subsets of \(K\) into subsets of \(K\). If for each finite subgraph \((A,\Gamma_A)\) there is a function \(\psi\) such that for each \(x\in A\), \(\psi(x)\in T(\psi[\Gamma_A(x)])\), then there is a function \(\phi\) such that for all \(x\in G\), \(\phi(x)\in T(\phi[\Gamma(x)])\). |
| 62: | \(C(\infty,< \aleph_{0})\): Every set of non-empty finite sets has a choice function. |
| 61: | \((\forall n\in\omega, n\ge 2\))\((C(\infty,n))\): For each \(n\in\omega\), \(n\ge 2\), every set of \(n\) element sets has a choice function. |
| 45-n: | If \(n\in\omega-\{0,1\}\), \(C(\infty,n)\): Every set of \(n\)-element sets has a choice function. |
| 33-n: | If \(n\in\omega-\{0,1\}\), \(C(LO,n)\): Every linearly ordered set of \(n\) element sets has a choice function. |
| 47-n: | If \(n\in\omega-\{0,1\}\), \(C(WO,n)\): Every well ordered collection of \(n\)-element sets has a choice function. |
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