Imagine you are the proposer. You will start at the top of your list and work your way down only if rejected. So you are guaranteed to the best you could possibly do.
Imagine
A ranks X,Y,Z in that order
B ranks Y,X,Z in that order
C ranks X,Y,Z
X, Y, Z rank A,B,C in that order.
A will propose to X and match.
B will propose to Y and match.
C will propose to X and get rejected.
C will propose to Y and get rejected.
C will propose to Z and match.
Y will never get a proposal from A. Z will never get a proposal from A or B.
(A,X),(B,Y) as the proposals is fine for the point being made. The issue is if we reverse which group is the proposer, we still end up with the same matching, because (A,X) strictly prefer each other.
The minimal example with 4 participants is if each person P has unique preferences and P's 1st choice has P as 2nd choice. So in this case just flip X:
A:XY
B:YX
X:BA
Y:AB
If A,B propose you get AX,BY. If X,Y propose you get BX,AY. Both are stable because the proposers are getting their first choice.
Not sure if you're referring to the actual definition. It's basically "no two participants from different pairs in the matching would prefer each other to their current matches." It seems an elegant definition to me.
This obviously doesn't factor in most of the complexity of romantic relationships: not knowing your own preferences, evolving preferences, incomplete/imperfect information in general, etc. Plus it assumes a global 1:1 matching across two categories, and people can of course be LGBT or choose to be single.
Imagine A ranks X,Y,Z in that order B ranks Y,X,Z in that order C ranks X,Y,Z
X, Y, Z rank A,B,C in that order.
A will propose to X and match. B will propose to Y and match. C will propose to X and get rejected. C will propose to Y and get rejected. C will propose to Z and match.
Y will never get a proposal from A. Z will never get a proposal from A or B.
Edit: I think I fixed it.