Some people may find this fascinating, and others may dismiss it as mathematical mumbo-jumbo. That's fine; most people treat the issue of unrestricted marriage from a purely emotional standpoint, and if you're one of those people then I don't expect a discussion based on mere logic to have much of an impact.
The Marriage Problem is defined as follows: two equal-sized sets of people, male and female, need to form pairs. It's easy to do -- pairings can be done completely randomly, and as long as each man gets assigned to one woman, you're done. The problem is that when these people have preferences as to who they get paired up with, the situation gets much more complicated.
For example, if M1 gets assigned to W1, but would prefer to be with W2, he will leave W1 if W2 is willing to pair up with him -- that is, if W2 prefers M1 over whoever her current mate is. As far as M1 is concerned, he is willing to leave his mate for another woman as long as there is a woman who he prefers over his mate, and that woman also prefers M1 over her mate. Got it?
A stable marriage is a set of matchings between these men and women such that, for any given man, there is no woman he prefers over his mate that also prefers him over her mate. The man may prefer other women over his mate, but as long as they don't want him it doesn't matter; he won't leave his mate if he doesn't have anyone else to go to.
As it turns out, it's pretty easy to form stable marriages. Essentially, you let all the pairings swap back and forth, and eventually an equilibrium state will be reached such that no one wants to change. That's an intuitive result, but it wasn't easy to prove that it works under every possible set of preferences. That is, no matter how convoluted the preferences of the two groups of people may be, there is guaranteed to be at least one stable marriage that will match the men to the women and not lead to any more swapping.
Among humans, we typically refer to this as "dating". People get to know each other and try each other out, and eventually commit. Unlike the mathematical model, the sets of men and women are not really finite; there's always a chance that you'll meet someone new that you haven't considered before, and you may prefer her to your wife, and she may be willing to take you. Nevertheless, stable marriages are theoretically guaranteed as long as the initial conditions are met.
The crucial initial condition is that there are two sets of people, and that no individual is willing to be matched with another individual in the same set. This is called a "binary partition" (divided in two), and without it the entire proof falls apart. In fact -- except in a few very specific instances with carefully crafted sets of preferences -- it's impossible to form a stable marriage among a single set of people.
[I tried to write a simple illustration of why, but you'll have to read the proofs yourself, I think. I've tried to keep this whole post as simple as possible, and I hope that hasn't led me to omit any important details.]
Homosexuality eliminates the two groups we normally deal with (men and women) and lumps everyone together into a single group. Without getting caught up on the terminology of "marriage", it has been mathematically proven that homosexual pairings are less stable than heterosexual pairings, wholly and simply due to the mate preferences of the people involved. Individual humans aren't as rigid as mathematical models, of course, but the aggregate behavior of real humans will closely match what is predicted by theory.
