Tackling difficult logic question I am stuck on this question.
Could somebody give some advice on how to tackle/solve this question?
I believe that with „always one“ the question means at least one. (otherwise it will not work)
Your help would be very appreciated.
Michal

For a big group of people we know that:
  If we select any four persons from the group, that there is always one of the four persons that will be friends with the other three persons.

Therefore, for any four persons selected from the group, will there always be one person that is friends will all people from the whole group?
 A: Assume the conclusion is false, then we find four people $A,B,C,D$ in the group for which all four of them have someone in the group who is not their friend. In particular we will find that $A$ and $B$ both have someone who is not their friend, denoted by $\bar{A}$ and $\bar{B}$ respectively. Consider the configuration $A,\bar{A},B,\bar{B}$. Now clearly here we don't find a person who is friends with the other $3$, contradicting the assumption.
Edit: If $\bar{A}$ and $\bar{B}$ are the same person, consider $A,C,\bar{A},\bar{C}$. (Where $\bar{C}$ is the "antifriend" of $C$). If also $\bar{C}=\bar{A}$ then $A,B,C,\bar{A}$ will do.
A: Here is a proof using induction.  For four people it is certainly true.  Now suppose you have a group of $n$ people and the claim is true for all groups of size less than $n$.  Select person $A$ at random and remove them from the group.  This group will have someone who is friends with everyone.  Set them aside with $A$ and call them $B$.  Do this two more times to get $C$ and $D$.  One of $A$, $B$, $C$, or $D$ will be friends with the three others.  If it's $A$ then $B$ is friends with $A$ and everyone else by assumption.  So, $B$ is friends with the group of $n$ people.  If it is $B$, $C$, or $D$ then they will be friends with the whole group of $n$ people.
