An Introduction to Large Numbers
Sidney Smith
Boise State University
April 2, 2015
Sidney Smith
An Introduction to Large Numbers
Boise State University
Ramsey Theory
Sidney Smith
An Introduction to Large Numbers
Boise State University
Ramsey Theory
Ramsey Theory is the mathematical study of when order begins to
arise within certain systems.
Sidney Smith
An Introduction to Large Numbers
Boise State University
Ramsey Theory
Ramsey Theory is the mathematical study of when order begins to
arise within certain systems.
For Example
Sidney Smith
An Introduction to Large Numbers
Boise State University
Ramsey Theory
Ramsey Theory is the mathematical study of when order begins to
arise within certain systems.
For Example
Image an n-dimensional hypercube where each vertex is connected
by one edge to each other vertex. Each of these edges are colored
one of two colors. When is it unavoidable that there is a complete
graph k4 that is one color and within one plane?
Sidney Smith
An Introduction to Large Numbers
Boise State University
Example
This is an example of a 3-dimensional hypercube, otherwise known
as a cube, having a k4 of one color, in one plane.
However it is avoidable to have this condition in the graph.
Sidney Smith
An Introduction to Large Numbers
Boise State University
Bounds Found
Sidney Smith
An Introduction to Large Numbers
Boise State University
Bounds Found
The lower bound found so far is 13.
Sidney Smith
An Introduction to Large Numbers
Boise State University
Bounds Found
The lower bound found so far is 13.
The upper bound was originally proven to be Graham’s Number.
Sidney Smith
An Introduction to Large Numbers
Boise State University
Finding Graham’s Number
In order to find Graham’s number we need to know Knuth’s
Up-Arrow Notation.
Sidney Smith
An Introduction to Large Numbers
Boise State University
Finding Graham’s Number
In order to find Graham’s number we need to know Knuth’s
Up-Arrow Notation.
This notation uses ↑ to describe large numbers
Sidney Smith
An Introduction to Large Numbers
Boise State University
Finding Graham’s Number
In order to find Graham’s number we need to know Knuth’s
Up-Arrow Notation.
This notation uses ↑ to describe large numbers
3 ↑ 3 = 33 = 27
Sidney Smith
An Introduction to Large Numbers
Boise State University
Finding Graham’s Number
In order to find Graham’s number we need to know Knuth’s
Up-Arrow Notation.
This notation uses ↑ to describe large numbers
3 ↑ 3 = 33 = 27
3
3 ↑↑ 3 = 33↑3 = 33 = 7625597484987
Sidney Smith
An Introduction to Large Numbers
Boise State University
Finding Graham’s Number
In order to find Graham’s number we need to know Knuth’s
Up-Arrow Notation.
This notation uses ↑ to describe large numbers
3 ↑ 3 = 33 = 27
3
3 ↑↑ 3 = 33↑3 = 33 = 7625597484987
3
3 ↑↑↑ 3 = 3 ↑↑ 3 ↑↑ 3 = 33
Sidney Smith
An Introduction to Large Numbers
.
..
This number has 3.6 trillion digits.
Boise State University
Finding Graham’s Number
In order to find Graham’s number we need to know Knuth’s
Up-Arrow Notation.
This notation uses ↑ to describe large numbers
3 ↑ 3 = 33 = 27
3
3 ↑↑ 3 = 33↑3 = 33 = 7625597484987
3
3 ↑↑↑ 3 = 3 ↑↑ 3 ↑↑ 3 = 33
.
..
This number has 3.6 trillion digits.
3 ↑↑↑↑ 3
Sidney Smith
An Introduction to Large Numbers
Boise State University
Finding Graham’s Number
In order to find Graham’s number we need to know Knuth’s
Up-Arrow Notation.
This notation uses ↑ to describe large numbers
3 ↑ 3 = 33 = 27
3
3 ↑↑ 3 = 33↑3 = 33 = 7625597484987
3
3 ↑↑↑ 3 = 3 ↑↑ 3 ↑↑ 3 = 33
.
..
This number has 3.6 trillion digits.
3 ↑↑↑↑ 3 more then the numbers of atoms in the universe by
an extremely large factor
Sidney Smith
An Introduction to Large Numbers
Boise State University
Put it in Perspective
To hopefully put this in perspective imagine a teaspoon of water.
Sidney Smith
An Introduction to Large Numbers
Boise State University
Put it in Perspective
To hopefully put this in perspective imagine a teaspoon of water.
That teaspoon of water has about
167,300,000,000,000,000,000,000 atoms in it.
Sidney Smith
An Introduction to Large Numbers
Boise State University
Put it in Perspective
To hopefully put this in perspective imagine a teaspoon of water.
That teaspoon of water has about
167,300,000,000,000,000,000,000 atoms in it.
Each gallon of water has 768 teaspoons in it.
Sidney Smith
An Introduction to Large Numbers
Boise State University
Put it in Perspective
To hopefully put this in perspective imagine a teaspoon of water.
That teaspoon of water has about
167,300,000,000,000,000,000,000 atoms in it.
Each gallon of water has 768 teaspoons in it.
The Pacific Ocean has over 187,000,000,000,000,000,000
gallons in it.
Multiplying all these together gets you the number of atoms in the
Pacific Ocean.
Sidney Smith
An Introduction to Large Numbers
Boise State University
Graham’s Number
In order to reach Graham’s Number we start with 3 ↑↑↑↑ 3.
Sidney Smith
An Introduction to Large Numbers
Boise State University
Graham’s Number
In order to reach Graham’s Number we start with 3 ↑↑↑↑ 3.
We will have this be equal to G1 .
Sidney Smith
An Introduction to Large Numbers
Boise State University
Graham’s Number
In order to reach Graham’s Number we start with 3 ↑↑↑↑ 3.
We will have this be equal to G1 .
G2 = 3 ↑↑ ... ↑↑ 3, Where the number of arrows is equal to G1 .
Sidney Smith
An Introduction to Large Numbers
Boise State University
Graham’s Number
In order to reach Graham’s Number we start with 3 ↑↑↑↑ 3.
We will have this be equal to G1 .
G2 = 3 ↑↑ ... ↑↑ 3, Where the number of arrows is equal to G1 .
Going to G64 yields Graham’s Number.
Sidney Smith
An Introduction to Large Numbers
Boise State University
Graham’s Number
In order to reach Graham’s Number we start with 3 ↑↑↑↑ 3.
We will have this be equal to G1 .
G2 = 3 ↑↑ ... ↑↑ 3, Where the number of arrows is equal to G1 .
Going to G64 yields Graham’s Number.
Graham’s Number ends with the last digits ...4195387
Sidney Smith
An Introduction to Large Numbers
Boise State University
Conclusion
In relation to the original Ramsey Theory question, Graham’s
Number has been proven to be the upper bound of the number of
dimensions required to be guaranteed to find a K4 graph of one
color in one plane in a n-dimensional hypercube.
Sidney Smith
An Introduction to Large Numbers
Boise State University