Form an undirected graph where the vertices are academics, and an edge connects academic X to academic Y if X has written a paper with Y. The Erdos number of X is the length of the shortest path in this graph connecting X with Erdos.
Erdos has Erdos number 0. Co-authors of Erdos have Erdos number 1. Einstein has Erdos number 2, since he wrote a paper with Ernst Straus, and Straus wrote many papers with Erdos.
The Extended Erdos Number applies to co-authors of Erdos. For People who have authored more than one paper with Erdos, their Erdos number is defined to be 1/# papers-co-authored.
Why people care about it?
Nobody seems to have a reasonable answer...
Who is Paul Erdos?
Paul Erdos was an Hungarian mathematician. He obtained his PhD from the University of Manchester and spent most of his efforts tackling "small" problems and conjectures related to graph theory, combinatorics, geometry and number theory.
He was one of the most prolific publishers of papers; and was also and indefatigable traveller.
Paul Erdös died on September 20, 1996.
At this time the number of people with Erdos number 2 or less is estimated to be over 4750, according to Professor Jerrold W. Grossman archives. These archives can be consulted via anonymous ftp at vela.acs.oakland.edu under the directory pub/math/erdos or on the Web at http://www.acs.oakland.edu/ grossman/erdoshp.html. At this time it contains a list of all co-authors of Erdos and their co-authors.
On this topic, he writes
Let E_1 be the subgraph of the collaboration graph induced by people with Erdos number 1. We found that E_1 has 451 vertices and 1145 edges. Furthermore, these collaborators tended to collaborate a lot, especially among themselves. They have an average of 19 other collaborators (standard deviation 21), and only seven of them collaborated with no one except Erdos. Four of them have over 100 co-authors. If we restrict our attention just to E_1, we still find a lot of joint work. Only 41 of these 451 people have collaborated with no other persons with Erdos number 1 (i.e., there are 41 isolated vertices in E_1), and E_1 has four components with two vertices each. The remaining 402 vertices in E_1 induce a connected subgraph. The average vertex degree in E_1 is 5, with a standard deviation of 6; and there are four vertices with degrees of 30 or higher. The largest clique in E_1 has seven vertices, but it should be noted that six of these people and Erdos have a joint seven-author paper. In addition, there are seven maximal 6-cliques, and 61 maximal 5-cliques. In all, 29 vertices in E_1 are involved in cliques of order 5 or larger. Finally, we computed that the diameter of E_1 is 11 and its radius is 6.Three quarters of the people with Erdos number 2 have only one co-author with Erdos number 1 (i.e., each such person has a unique path of length 2 to p). However, their mean number of Erdos number 1 co-authors is 1.5, with a standard deviation of 1.1, and the count ranges as high as 13.
Folklore has it that most active researchers have a finite, and fairly small, Erdos number. For supporting evidence, we verified that all the Fields and Nevanlinna prize winners during the past three cycles (1986--1994) are indeed in the Erdos component, with Erdos number at most 9. Since this group includes people working in theoretical physics, one can conjecture that most physicists are also in the Erdos component, as are, therefore, most scientists in general. The large number of applications of graph theory to the social sciences might also lead one to suspect that many researchers in other academic areas are included as well. We close with two open questions about C, restricted to mathematicians, that such musings suggest, with no hope that either will ever be answered satisfactorily: What is the diameter of the Erdos component, and what is the order of the second largest component?
Caspar Goffman. And what is your Erdos number? American Mathematical Monthly, v. 76 (1969), p. 791.
Tom Odda (alias for Ronald Graham) On Properties of a Well- Known Graph, or, What is Your Ramsey Number? Topics in Graph Theory (New York, 1977), pp. [166-172].