# Party Numbers: Deploying a Host of Computers to Sort out a Mathematical Puzzle

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You're one of six people at a dinner party You look across the table at your five companions. It should come as no surprise to you that the dinner party includes either a group of at least three people who all know one another or a group of at least three people who don't know one another.

The reason for this certainty lies in a mathematical proof that any gathering of six people will always automatically include one or the other of these two groupings. No such guarantee is possible when five or fewer people are present.

This result stems from a branch of pure mathematics known as Ramsey theory, which concerns the existence of highly regular patterns in any large set of randomly selected numbers, points, or objects. Given enough stars, for example, it's not difficult to find groups of stars that very nearly form a straight line, a rectangle, or even a dipper.

Similarly, any sufficiently long sequence of numbers generated by the roll of a die will inevitably display certain regularities. Ramsey theorists try to work out just how many stars, numbers, or party goers are required to guarantee the presence of a certain pattern.

The party puzzle typifies the sorts of problems that Ramsey theory tackles: What is the minimum number of guests that must be invited so that either at least x guests will know one another or at least y guests won't? The resulting minimum number--which equals 6 when x is 3 and y is 3--is called a Ramsey number.

The search for Ramsey numbers belongs to the realm of pure mathematics. So far, there are no direct practical applications. Nonetheless, the techniques developed to look for Ramsey numbers may themselves prove useful one day in both mathematics and computer science.

Determining the Ramsey numbers associated with various combinations of acquaintances and strangers has proved extraordinarily difficult (see table). Now, two researchers have finally shown that 25 is the minimum number of guests needed to guarantee that a party includes a group of at least four people who all know one another or a group of at least five people who are strangers to one another. Using as many as 110 desktop computers at one time to sift through various possibilities, Stanislaw P. Radziszowski of the Rochester (N.Y) Institute of Technology and Brendan D. McKay of the Australian National University in Canberra spent nearly three years establishing this result.

"This was the smallest unsolved Ramsey number," Radziszowski says. "It will quite probably be the last one solved for many, many years."

Ramsey theory originated in the work of Frank Plumpton Ramsey, a mathematician at the University of Cambridge in England who was keenly interested in mathematical logic, philosophy, and economics. Although only 26 years old when he died in 1930, Ramsey left behind a rich legacy of mathematical results that continue to intrigue mathematicians.

"He is especially noted for the remarkable originality of his work, which was in many cases not appreciated until long afterwards," Radziszowski says. The party puzzle represents a special case of a theorem that appeared in one of Ramsey's papers on mathematical logic. This theorem states that if the number of objects in a set is sufficiently large and if each pair of objects has one of a number of relations, there exists a subset containing a certain number of objects where each pair has the same relation (that is, all the members of the subset are mutual acquaintances or strangers).

That's like saying that complete disorder is impossible, remarks mathematician Ronald L. Graham of AT&T Bell Laboratories in Murray Hill, N.J. "Somehow, no matter how chaotic something looks, deep within it is something smaller that's more structured."

In the 1930s, Hungarian mathematician Paul Erdos, who pioneered many of the key ideas extending Ramsey's theorem, started thinking about the question of exactly how large a set must be to guarantee the presence of a certain subset. …