Ramsey Theory &
The World's Largest Number



The world's largest number, according to the Guiness Book of Records, is Graham's number. The number is named for Ronald L Graham, juggler, acrobat, and mathematician. Of course every good math student knows there can be no "largest" number. Actually the record book lists it as the largest number ever used in a proof. The number is incomprehensibly large, beyond images like the number of grains of sand needed to fill the universe or any such comparison. It can not even be described with conventional numerical techniques. To illustrate the undescribability of its size, David Wells wrote in The Penguin Dictionary of Curious and Interesting Numbers, "If all the material in the universe was turned into pen and ink it would not be enough to write the number down."
To write the number a special notation invented by Donald Knuth is required, and even then it grows out of reason ... so I will once more quote David Wells to insure I get the symbols straight...
3^3 means '3 cubed', as it often does in computer printouts.
3^^3 means 3^(3^3), or 3^27, which is already quite large: 3^27 = 7,625,597,484,987...
3^^^3 = 3^^(3^^3), however, is 3^^7,625,597,484,987 = 3^(7,625,597,484,987^7,625,597,484,987), which makes a tower of exponents 7,625,597,484,987 layers high.
3^^^^3 = 3^^^(3^^^3), of course. Even the tower of exponents is now unimaginably large in our usual notation, but Graham's number only starts here.
Consider the number 3^^^...^^^3 in which there are 3^^^^3 arrows. A largish number!
Next construct the number 3^^^ ... ^^^3 where the number of arrows is the previous 3^^^...^^^3 number. An incredible, ungraspable number! Yet we are only two steps away from the original ginormous 3^^^^3.
Now continue this process, making the number of arrows in 3^^^ ... ^^^3 equal to the number at the previous step, until you are 63 steps, yes, sixty-three, steps from 3^^^^3. That is Graham's number.

You may well ask yourself, what kind of a proof would involve a number so large? The answer takes us to a description of an area of combinatorics called Ramsey Theory, named after Frank P Ramsey, a brillant young mathematician whose life was cut short by Jaundice in 1930 at the young age of twenty-six. The basic question in Ramsey theory is "What is the smallest set that MUST contain a given subset?" A simple example would be how many people are necessary to have at least two of one sex or the other. The answer in this case is simple, with a set of three people we know that at least two will be of the same gender.

Graham's number comes about as the upper limit of the solution of such a problem. Here is the problem that Graham solved.

Take any number of people, list every possible committee that can be formed from them, and consider every possible pair of committees. How many people must be in the original group so that no matter how the assignments are made, there will be four committees in which all the pairs fall in the same group, and all the people belong to an even number of committees.

While the unthinkably large upper boundary to the answer is given by Graham's number, nobody, including Graham himself, believe the answer is nearly so large. Most Ramsey Theory experts, in fact, suspect that the answer is SIX people!!!!!

Arthur Michael Ramsey, Frank Ramsey's one year younger brother, was Archbishop of Canterbury from 1961 to 1974.

Frank Ramsey is buried in the Burial Ground of Parish of the Ascension in Cambridge, UK. An image of his gravestone is here