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Notes: Chapter 2
Section 2.3: The Pigeonhole Principle
Definition 1. If more than n · k objects are placed into n categories,
then there exists a category with more than k objects in it.
Example 1. A drawer with 8 black socks and 11 blue socks. It’s dark
and you can’t tell them apart. How many must you take to guarantee
2 of the same color? 2 black? 2 blue?
Example 2. A society of friends. Given a set S of people, there must
be 2 people with the same number of friends.
Example 3. There are 30 balls with the numbers 1 through 30 written
on them. If 18 balls are drawn, there must be a pair whose sum is 35.
Proof. The categories are the pairs that have a sum of is 35, plus the
single balls that are not part of any pair, 1,2,3,4.
Example 4. Let S ⊆ [3n], with |S| = 2n + 1. Then there will be 3
consecutive numbers.
Proof. To avoid consecutive, must have one only in each of the categories: C1 = {1, 2, 3}, C2 = {4, 5, 6}, C3 = {7, 8, 9}, . . . , Cn = {3n −
2, 3n − 1, 3n}. But there are n categories and more than 2n objects, so
one category must contain 3 objects.