# The “Marriage Theorem” Theorem

The Marriage Theorem is a beautiful piece of mathematics, proved in the 1930s by mathematician Philip Hall. Suppose we have a number of men and the same number of women. Each man is happy to marry some (but perhaps not all) of the women, and similarly for each woman. The question is, can we pair up all the men and women so that everyone is happily married?

Obviously this will be impossible if too many people are too fussy. We’ll definitely require, for example, each woman to be happy to marry at least one man. Similarly, if we take any pair of women then there’s no hope if those two women are both just keen on the one and same man. More generally, we can take any collection W the women, and then we can consider the collection M of men who are acceptable to at least one of those women. The marriage condition states that, no matter the collection W, the corresponding collection M is at least as large as W.

If the marriage condition is not satisfied then there’s definitely no hope of happily marrying everyone off. (If the condition fails for some W then there simply aren’t enough acceptable men for all the women in W.) The Marriage Theorem is the surprising result that the marriage condition is all we need to check; if the marriage condition is satisfied then everyone can be happily married.

That’s all well and good. It’s a beautiful theorem, and you can check out a very nice proof at (no pun intended) cut-the-knot. This, however, is a blog about mathematical crap. So, where’s the crap? For that, we head off to Sydney’s University of New South Wales.

It appears that a lecturer at UNSW who has been teaching the Marriage Theorem has requested that students not refer to the theorem by that name, because of the “homophobic implications”; use of the term in student work was apparently marked as “offensive”. How do we know this? Because one of the affected students went on Sky News to tell the story.

But, at least we have a new theorem:

The “Marriage Theorem” Theorem

a) Any mathematician who whines to her students about the title “Marriage Theorem” is a trouble-making clown with way too much time on her hands.

b) Any student who whines about the mathematician in (a) to a poisonously unprincipled pseudonews network is a troublemaking clown with way too much time on his hands.

Proofs: Trivial.

## 2 Replies to “The “Marriage Theorem” Theorem”

1. Anonymous says:

I went out with a gal who was a math prof (compsci, but basically math). She mentioned this and said that the pairings could be optimal for the women or the men. (I don’t know why/how, she was just chatting, not proving.) She did say that some grad schools (maybe top medical schools?) did a pairing algorithm to sort students amongst top colleges. And that the pairing could be biased to the students optimizing or the schools…and that surprise surprise, it was towards the schools.

1. marty says:

Yes, it’s a famous theorem, by David Gale and Lloyd Shapely. Burkard and I (and many others) wrote a column about it, which we included in our Dingo collection.