Respuesta :
a group with at least n k 1 people such that no two members of the group are friends,We will prove this statement by inductive hypothesis.
Base Case: If n=k+1, then we can easily find a group with nk1 people such that no two members of the group are friends. We simply select the k+1 people in the room, excluding Alex. Since Alex has k friends, and no other person in the room has more than k friends, it follows that none of these k+1 people are friends.
Inductive Step: Assume that the statement is true for n=k+1. Now we must show that it is true for n=k+2.
We can select a group of k+1 people from the room, again excluding Alex. By the inductive hypothesis, we know that no two members of this group are friends. Now, we select the remaining person in the room, and add them to the group. This person cannot be friends with any of the members of the group, since Alex has k friends and no other person in the room has more than k friends. Therefore, the group we have formed has at least nk1 people, and no two members of the group are friends.
Therefore, by induction, we have proved that we can find a group with at least nk1 people such that no two members of the group are friends.
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