I’ve been working on this question for a few days now and I’m getting very confused and unsure how to progress. I’d like to show that if there are $n$ boys and $n$ girls in a village such that every group of $k$ boys knows at least $k$ girls, that I can marry each girl with a boy she knows. For $n=1$ there is trivially one possibility for matching boy with girl. For $n=2$ I’ve shown using bipartite graphs that there are $7$ options for knowing each other and for each possibility I can find suitable marriage arrangements.
The problem starts with $n=3$. There are so many possible graphs to draw that you need to start proving it more rigorously in writing. I considered the case in which the fewest no. of girls any of the boy knows is $1,2,3$ separately and found it worked in each case.
Now for $n=4$ it is getting very hard. If it is very hard for $n=4$ then I’m not sure where to even start the general $n=k$ case to prove it by induction.
I thought about trying to write an algorithm to turn a suitable ‘knowledge of each other’ graph into a ‘marriage graph’.
I thought: consider the boys who know only one girl and pair them up. These girls will be distinct because all these boys must know as many girls. Then consider the boys who know two girls and so on. But this is very longwinded and I find myself lost regularly.
Would someone be able to tell me where I may be going wrong and help me start going in the right direction?
The statement you want to prove is essentially Hall's marriage theorem. With the conditions you state, this theorem states that it is always possible to match the boys to the girls, and since the numbers on both sides are equal, no girl is left unpaired.