Problem 3. Suppose that there is a group of n people, where n 2 2. Prove that there are at least two people in the group with the same number of friends. Some notes: every two people are either friends or are not, if a is a friend of b then b is a friend of a, and no person is their own friend - in other words, friendship is a symmetric, irreflexive relation on this set of n peo- ple. (Hints: use the pigeonhole principle, start by explaining why it's not possible for one per- son to have 0 friends and another person to have n-1 friends.)

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Problem 3. Suppose that there is a group of n people, where n 2 2. Prove that there are at
least two people in the group with the same number of friends. Some notes: every two people
are either friends or are not, if a is a friend of b then b is a friend of a, and no person is their
own friend – in other words, friendship is a symmetric, irreflexive relation on this set of n peo-
ple. (Hints: use the pigeonhole principle, start by explaining why it's not possible for one per-
son to have 0 friends and another person to have n-1 friends.)
Transcribed Image Text:Problem 3. Suppose that there is a group of n people, where n 2 2. Prove that there are at least two people in the group with the same number of friends. Some notes: every two people are either friends or are not, if a is a friend of b then b is a friend of a, and no person is their own friend – in other words, friendship is a symmetric, irreflexive relation on this set of n peo- ple. (Hints: use the pigeonhole principle, start by explaining why it's not possible for one per- son to have 0 friends and another person to have n-1 friends.)
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