Pigeon Hole Principle

The basic principle is very simple:

If there are pigeons and m < n holes to put those pigeons in, then at least two pigeons need to be put in the same hole.

Even this simple principle can help prove some very surprising results.

Example-1

Imagine a party with guests. Assume that everyone in the party knows at least one other person but is a stranger to at least one other person. There are at least two people who know the same number of people.

Example-2

Zip is a commonly used compression algorithm. We know that zip can compress most files. By pigeon hole principle we can conclude that there is at least one file that zip will not compress.

Example-3

Sequential circuit output:

A Boolean circuit has three LEDs at its output that can either glow red or green depending on the input. It has 4 Boolean input wires and for each input, it has a specific patterns for output LEDs. Show that there are at least two inputs for which the output pattern will be the same.

Extended Pigeon Hole Principle

If there are m times n +1 objects to be distributed in bins, then at least one bin has more than objects.

Example-1

Let's say this class has students attending on average. I can confidently say that at least two of you will have a birthday that falls on the same week.

In fact, at least of you will have a birthday that falls on the same month. How can I say these statements without knowing your birthdays? :-)

Assuming all of you hail from somewhere in the USA, I happen to know that at least two of you are from the same state. :-)

Example-2

Sequential circuit output: A Boolean circuit has three LEDs at its output that can either glow red or green depending on the input. It has 5 Boolean input wires and for each input, it has a specific patterns for output LEDs.

We can now show that there are at least inputs that produce the same output.

Pigeon Hole Principle to Prove Properties of Numbers

Theorem Whenever distinct points are placed inside a 1 times 1 square at least two will be within a distance of less than $frac{1}{sqrt{2}}$ .

Proof We need to draw a picture of a square and divide it into four equal squares of size $frac{1}{2} times frac{1}{2}$ each. By PH principle, at least two of the points will be on the same small square. The longest distance these two points can be apart inside the small square is the length of its diagonal.

Theorem Whenever distinct points are placed on a circle with center , there will be at least two points P,Q with the angle PCQ being acute.

Proof Again divide the circle into four quadrants. Two of the points have to lie on the same quadrant by pigeon hole principle.

Theorem Whenever five points are placed on the surface of a sphere, the sphere can be cut into two halves (hemispheres) such that four points are on one half.

Ramsey's Theorem

Ramsey's theorem is a very cool fundamental fact of combinatorics. It has many (equivalent) flavors:

In any class of six or more people, there are at least three mutual friends or three mutual strangers.
Take 6 points that form the corners of a 6 sided figure and between every two points either draw a red line or a green line. Then there has to be a red triangle or a green triangle.
Any graph with nodes has a -clique or a -independent set.
Ramsey Number (3,3) = 6.
and so on.

We will prove this theorem in class.