The Pigeonhole Principle

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Jonah Logan
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1 The Pigeonhole Principle

2 Some Questions Does there have to be two trees on Earth with the same number of leaves? How large of a set of distinct integers between 1 and 200 is needed to assure that two numbers in the set have a common divisor? How large a set of distinct integers between 1 and n to assure that the set contains a subset of five equally spaced integers a 1, a 2, a 3, a 4, a 5 ; that is a 2 a 1 = a 3 a 2 = a 4 a 3 = a 5 a 4? Given a positive integer k, how large a group of people is needed to ensure that either there exists a subset of k people in the group that know each other or a subset of k people none of whom know each other?

3 The Pigeonhole Principle Theorem The Pigeonhole Principle If k + 1 pigeons are placed into k pigeonholes then at least one of the pigeonholes contains two or more pigeons.

4 The Pigeonhole Principle Theorem The Pigeonhole Principle If k + 1 pigeons are placed into k pigeonholes then at least one of the pigeonholes contains two or more pigeons. A more general statement of the theorem: Theorem If m pigeons are to be placed into k pigeonholes, then at least one of the pigeonholes will contain more than m 1 pigeons. k

5 Basic s How many people are needed to have two people born on the same day of the week? If a person owns 5 mutual funds holding a total of 56 stocks, what can you say about the number of stocks in the largest fund? If SSU has 6000 undergrads and there is at least one from each of the 50 states, then there is at least one state with at least 120 students from that state.

6 A Little Bit Tougher If you draw 5 points on the surface of an orange with a Sharpie, then there is a way to cut the orange in half so that four of the points will lie on the same hemisphere. (Suppose a point exactly on a cut lies in both hemispheres.)

7 A Little More To Think About Prove that at a party with at least two people, there are at least two people who know the same number of other people.

8 Another Points in a Region Consider the region bound by a regular hexagon where each side has length 1 unit. Show that if any seven points are chosen in this region then two of them must be no further apart than 1 unit.

9 Another Medium Prove that if seven distinct numbers are selected from {1, 2,... 11}, then some two of these numbers sum to 12.

10 And Another Medium Prove that if four points are selected from the interior of a unit circle, then there are two points whose distance apart is less than 2.

11 Would You Play This Game? Twenty disks numbered 1 through 20 are placed face down on a table. Disks are selected one at a time and turned over until 10 disks have been chosen. If two of the disks add up to 21, the player loses. Is it possible to win the game?

12 A Number Theory Prove that any collection of eight distinct integers contains distinct integers x and y such that x y is a multiple of 7.

13 Subsequences Prove that any sequence of n integers must contain a subsequence whose sum is divisible by n.

14 A Tougher A coach wants to hold 31 practices over a 21 day preseason. There will be at least one practice per day. Is there a stretch of consecutive days where there are exactly 10 practices?

15 Erd.. os and Szekeres (1935) Theorem Given a sequence of n distinct integers, either there is an increasing subsequence of n + 1 terms or a decreasing subsequence of n + 1 terms.

16 Erd.. os and Szekeres (1935) Theorem Given a sequence of n distinct integers, either there is an increasing subsequence of n + 1 terms or a decreasing subsequence of n + 1 terms. n is a tight lower bound. Consider, for n = 3, the sequence of n 2 distinct integers. 3, 2, 1, 6, 5, 4, 9, 8, 7

17 Ramsey Theory Assume that among six persons, each pair of persons are either friends or enemies. Then there are either three persons who are mutual friends or three persons who are mutual enemies.

18 Ramsey Numbers Theorem Let p, q 2. A positive integer N has the (p, q) Ramsey property if the following holds: Given any set S of N elements, if we divide the 2-element subsets of S into two classes X and Y, then either 1 there is a p-element subset of S all of whose two element subsets are in X, or 2 there is a q-element subset of S, all of whose 2-element subsets are in Y

19 of Ramsey Theory Show that 5 does not have the (3, 3) Ramsey property.

20 Ramsey Theory Frank Ramsey, British Mathematician Worked primarily in logic Studied a problem that seemed to have no order, but the situations always had a certain amount of order Invented Ramsey Theory Presented results in paper in 1928 at the London Mathematical Society Died at 28 from complications due to a liver condition at 26 before it was published

21 What is Ramsey Theory? Graph Theory Definition The Ramsey number is the positive integer N = R(n, m) such that a graph on N vertices must contain a complete subgraph on n vertices in one color or a complete subgraph on m vertices in the other color. We often think of the colors as red and blue, although it is arbitrary which color we assign to n and which to m.

22 s What is R(2, 3)?

23 Getting to R(2, n) What is R(2, 4)? What is R(2, n) for n 2? What is R(n, 2)?

24 Bigger Parameters What is R(3, 3)? What is R(3, 4)?

25 Known Ramsey numbers R(4, 4) = 18 R(5, 3) = 14 R(6, 3) = 18 R(7, 3) = R(5, 5) 55

12. 6 jokes are minimal.

12. 6 jokes are minimal. Pigeonhole Principle Pigeonhole Principle: When you organize n things into k categories, one of the categories has at least n/k things in it. Proof: If each category had fewer than n/k things in it then