# Erdős number

The Erdős number, honouring the late Hungarian mathematician Paul Erdős, one of the most prolific writers of mathematical papers, is a way of describing the "collaborative distance", in regard to mathematical papers, between an author and Erdős.

An author's Erdős number is defined inductively as follows:

• Paul Erdős has an Erdős number of zero.
• The Erdős number of author M is one plus the minimum among the Erdős numbers of all the authors with whom M coauthored a mathematical paper.

Erdős wrote around 1500 mathematical articles in his lifetime, mostly co-authored. He had 509 direct collaborators; these are the people with Erdős number 1. The people who have collaborated with them (but not with Erdős himself) have an Erdős number of 2 (6,984 people), those who have collaborated with people who have an Erdős number of 2 (but not with Erdős or anyone with an Erdős number of 1) have an Erdős number of 3, and so forth.

Erdős numbers have been a part of the folklore of mathematicians throughout the world for many years. Amongst all working mathematicians at the turn of the millennium, the numbers range up to 15, but the average is less than 5, and almost everyone with a finite Erdős number has a number less than 8.

According to Alex Lopez-Ortiz, all the Fields and Nevanlinna prize winners during the three cycles in 1986 to 1994 have Erdős number at most 9.

The Bacon number is an application of the same idea to the movie industry, connecting actors that appeared in a film together.

Jerry Grossman, Marc Lipman, and Eddie Cheng have been looking at some questions in pure graph theory motivated by these collaboration graphs.

Also, Michael Barr suggests "rational Erdős numbers", generalizing the idea that a person who has written p joint papers with Erdős should be assigned Erdős number 1/p. From the collaboration multigraph of the second kind (although he also has a way to deal with the case of the first kind) -- with one edge between two mathematicians for each joint paper they have produced -- form an electrical network with a one-ohm resistor on each edge. The total resistance between two nodes tells how "close" these two nodes are.

On April 20 2004 Bill Tozier, a researcher with Erdős number 4, offered the chance for collaboration to attain an Erdős number 5 in an auction on eBay. The final bid was \$1,031.

Another eBay auction offered an Erdős number of 2 for a prospective paper to be submitted for publication to Chance (a magazine of the American Statistical Association) about skill in the World Series of Poker and the World Poker Tour. It closed on 22 July 2004 with a winning bid of \$127.40. This is noteworthy because with the exception of a few co-authored articles to be published posthumously, 2 is the minimum number that can now be achieved.

It is jokingly said that Baseball Hall of Famer Hank Aaron has an Erdős number of 1 because they both autographed the same baseball when Emory University awarded them honorary degrees on the same day.