The town of Friendville has an interesting property. Given any two people in the town, they either know each other or they don't. If they don't know each other, then they can be introduced to each other. One single introduction will work for both people. That is, "Tom this is Phil, Phil this is Tom" counts as one introduction.
The other interesting property that this town has, is that if any group of n people get together, the number of introductions that must be made in order that everyone in the group knows everyone else is at most n-1.
Problem: Prove that the town can be divided into two groups (A and B) such that everyone in group A knows each other, and everyone in group B knows each other.
(In reply to re: Possible solution
by Steve Herman)
You are right.
In my quest to simplify as much as possible, I have over-simplified.
'B does not know A' says nothing about 'B does not know C', and 'D does not know A, or D does not know B', says nothing about 'D knows both A and B.'
As you correctly state, the method therefore seems to require the creation of additional residual 'holding' classes, something I was anxious to avoid, even though it is easy to see that they must reduce by the sort of reasoning that you've already outlined.
Edited on June 15, 2013, 11:47 am
Posted by broll
on 2013-06-15 11:46:30