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. Erdős is pronounced as IPA /ɛrdøːʃ/.

Definition
In order to be assigned an Erdős number, an author must co-write a mathematical paper with an author with a finite Erdős number. Paul Erdős has an Erdős number of zero. If the lowest Erdős number of a coauthor is X, then the author's Erdős number is X + 1.

Erdős wrote around 1500 mathematical articles in his lifetime, mostly co-written. He had 504 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. A person with no such coauthorship chain connecting to Erdős has an undefined (or infinite) Erdős number.

The Erdős number was most likely first defined by Casper Goffman, an analyst whose own Erdős number is 1. Goffman published his observations about Erdős's prolific collaboration in a 1969 article entitled "And what is your Erdős number?"

Impact
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 who have a finite Erdős number, the numbers range up to 15, the median is 5, the average Erdős number is 4.65 (according to the Erdős Number Project);  and almost everyone with a finite Erdős number has a number less than 8. Due to the very high frequency of interdisciplinary collaboration in science today, very large numbers of non-mathematicians in many other fields of science also have finite Erdős numbers. In biomedical research, for instance, it is common for statisticians to be among the authors of publications, and many statisticians can be linked via Tukey to Erdős. 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.

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.

Earlier mathematicians published fewer papers than modern ones, and more rarely published jointly-written papers. The earliest person known to have a finite positive Erdős number is either Richard Dedekind (born 1831, Erdős number 7) or Georg Frobenius (born 1849, Erdős number 3), depending on the standard of publication eligibility. It seems that older historic figures such as Leonhard Euler do not have finite Erdős numbers.

Outside mathematics
The Bacon number (as in the game Six Degrees of Kevin Bacon) is an application of the same idea to the movie industry, connecting actors that appeared in a film together to the actor Kevin Bacon. A small number of people are connected to both Erdos and Bacon and thus have a finite Erdos-Bacon number.

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, though apparently the winning bidder had no intention to pay. The winner (who already had an Erdős number of 3) considered it a "mockery", and said "papers have to be worked and earned, not sold, auctioned or bought".

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-written articles to be published posthumously, 2 is the lowest number that can now be achieved.

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