What Is Your Erdös Number?

I have purchased, last weekend, Paul Hoffman's book The Man Who Only Loved Numbers. This is a biography of the great and equally eccentric number theorist of all times Paul Érdös.

The book has been published in 1998, two years after the death of Érdös in Warsaw. The first 25% of Hoffman's book is interesting and entertaining at the same time. It tell the story of a man who devoted his whole life to Mathematics, who travelled from country to country carrying all his belongings in a briefcase and a bag.

Érdös, once stated that life is about conjecturing and proving theorems. He co-authered about 1,500 papers with 507 mathematicians. He was only beaten by Leonhard Euler in his publications count.

Due to this very prolific carreer, some mathematicians defined the so-called Érdös number. This number is defined as follows. The Érdös number of Érdös himself is 0. Every mathematician of the 507 who co-authered a paper with him has an Érdös 1. This goes inductively as follows: a person P has an Érdös of n if and only if all his collaborators have an Érdös greater than or equal to n-1, and at least one of those collaborators has an Érdös of n-1.

Another way to define Érdös number is to resort to Graph Theory concepts. Let C be a graph where every vertex is a mathematician. Two vertices v1 and v2 are connected by an edge if and only if v1 and v2 co-authered a research paper together. Then, the Érdös number of a vertex v is the distance between v and Érdös. The distance between two vertices in a graph being defined as the length of one of the shortest paths between these two vertices. If no path connects two vertices, then the distance between them in infinite.

Oakland University maintains a web site The Érdös Number Project that provides the names of the mathematicians with an Érdös of 1 or 2 (there are about 10000 lucky members in this category). This project also provides some statistics about these numbers. For instance, the average Érdös number is 4.65 and the median is equal to 5.

The American Mathematical Society offers a web application that can be used to calculate one's Érdös number (actually the application can calculate the distance between arbitrary pairs of researchers). I have used this application and found that my Érdös number is 4. Here is the path that connects me to Érdös:

Iskander Kort --> Denis Trystram --> Gerhard J. Woeginger --> János Komlós --> Paul Erdős.

On a final note, we will be celebrating on March 26th 2013, the 100th anniversary of Paul Érdös. I can wait to see what will be Google's doodle that day.