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I am to hold a presentation on marriage problems in a few weeks and I thought it would be nice to include a few consequences (or, preferably, equivalences) of Philip Hall's marriage theorem and Marshall Hall Jr.'s marriage theorem. I was able to find some things on wikipedia (https://en.wikipedia.org/wiki/Hall%27s_marriage_theorem), but I am sure there are many other interesting versions of the theorems. I especially could not find much about Marshall Hall Jr.'s variant of the theorem.

To be more precise I would like to know:

  1. What are some examples of important results implied by (or equivalent to) by either of the theorems?

and, for those with a background in reverse mathematics:

  1. Is this implication (or equivalence) provable in $\mathsf{RCA}_0$?

Even if there is no known proof of the second question, I appreciate any intuition.

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