MACM 101 Lecture 23: Lecture 23 Part 2_ Pigeonhole Principle

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If for 1 < j < i < 28 we have xj + 15 = xi, then from the start of day j + 1 to the end of day i, Herbert will play exactly 15 sets of tennis. Assume that in a group of six people, each two individuals are either friends or enemies. Show that there are either three mutual friends or three mutual enemies in the group. Let a be one of the six people. Of the five other people in the group, there are either three or more who are friends of a, or three or more who are enemies of a. Indeed, by the generalized pigeonhole principle, when 5 objects are divided into 2 sets, one of the sets contains at least 5/2 = 3 objects. In the former case, suppose that b, c, and d are friends of a.

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