Groups, Graphs, and Paul Erdos

Many people have the impression that mathematical research is largely a solitary pursuit. They imagine a mathematician squirreled away in some dingy garret, lonely wilderness cabin, or sparsely furnished cubicle, oblivious to everyday concerns and focused on a single problem, scribbling inscrutable equations across scraps of paper, and thinking long and hard before emerging with a eureka and a proof.

The dramatic announcement in 1993 by Andrew Wiles that he had proved Fermat's Last Theorem appeared to belong to this category of discovery. He had virtually separated himself from the rest of the mathematical community for nearly eight years to work on this problem. Only a select few were aware of what he was trying to accomplish.

Yet, Wiles had relied heavily on the previous work of other mathematicians who had tackled the same problem. He had occasionally tested his ideas on a few trusted experts in areas of mathematics relevant to his approach. And when reviewers later discovered a flaw in his original chain of logic, he obtained help from one of his former graduate students, Richard Taylor, to fill in the gap and complete the proof.

At the same time, the relative isolation that Wiles sought is certainly not the rule in mathematical research. Doing mathematics is really a remarkably social process. The abundance of meetings, conferences, workshops, colloquia, seminars, and other gatherings of mathematicians attests to the importance of collaboration. Electronic communication speeds and facilitates such interaction.

Paul Erdos, perhaps more than any other mathematician in modern times, has epitomized the strength and breadth of mathematical collaboration. Born more than 80 years ago in Hungary, Erdos has traveled the world for decades in search of new mathematics and new collaborators.

Over the years, Erdos has written hundreds of research papers on a wide range of mathematical topics. What's especially astonishing is the extent to which he has worked with other mathematicians to produce joint papers. Indeed, his efforts have become legendary in mathematical circles, and mathematicians have taken a characteristically mathematical way of describing them -- by inventing a new quantity called an Erdos number.

Mathematicians assign Erdos the number 0. People who have coauthored a paper with him are given the number 1. As of May 1996, there were 462 such coauthors. Another 4,566 people have the number 2 because they wrote a paper not with Erdos himself but with someone who wrote a paper with Erdos. The Erdos number 3 goes to anyone who has collaborated with someone who has written a paper with someone who coauthored a paper with Erdos, and so on.

Thus, any person not yet assigned an Erdos number who has written a joint mathematical paper with a person having an Erdos number n earns the Erdos number n + 1. Anyone left out of this assignment process has the Erdos number infinity.

This scorekeeping system started years ago, but Jerrold W. Grossman at Oakland University in Rochester, Mich., has become the compiler and guardian of the lists of collaborators. Working with Patrick D.F. Ion, an editor at Mathematical Reviews in Ann Arbor, Mich., he continues to update, correct, and expand his files on Erdos numbers.

"Over a span of more than 60 years, Paul Erdos has taken the art of collaborative research in mathematics to heights never before achieved," Grossman remarks. Grossman's lists, in turn, provide fascinating glimpses of mathematicians and how they choose to do mathematical research.

Fifty-six years ago, when Mathematical Reviews first started compiling a record of all published mathematical work, more than 90 percent of papers were solo works. Now, scarcely more than half are individual efforts, and the fraction of two-author papers has risen from less than 10 percent to about one-third.

Moreover, in 1940, very few papers had three authors, let alone four or more. "Now, about 10 percent of all papers in the mathematical sciences have three or more authors, including about 2 percent with four or more," Grossman says.

One can also look at mathematical collaborations as a graph -- an array of points connected by lines. Every mathematician is represented as a point, with the Erdos point somewhere near the center. Any two mathematicians who have collaborated on a paper are joined by a line. The result is monstrous tangle that snares nearly all mathematicians, with tentacles reaching into computer science, the physical and biological sciences, economics, and even the social sciences.

By determining the smallest number of edges linking any person to Erdos in this enormous collaboration graph, one can determine that person's Erdos number. Albert Einsten, for instance, had the Erdos number 2. Andrew Wiles has an Erdos number of at most 4.

Grossman's lists and statistics are fun to explore. Mathematicians can try to work out their own Erdos numbers, just to see where they fit into the webby world of mathematics. Nonmathematicians can see the tangle of human relationships that play a key role in advancing mathematics.

References:

Albers, Donald J., and G.L. Alexanderson, editors. 1985. Mathematical People: Profiles and Interviews. Boston: Birkhauser.

Cipra, Barry. 1996. What's Happening in the Mathematical Sciences, 1995-1996. Providence, R.I.: American Mathematical Society.

Grossman, Jerrold W. 1996. Paul Erdos: The master of collaboration (preprint).

Hoffman, Paul. 1987. The man who loves only numbers. The Atlantic Monthly 260 (Nov.): 60-74.

Odda, Tom [=Ronald L. Graham]. 1979. On properties of a well-known graph, or what is your Ramsey number? Annals of the New York Academy of Science 328: 166-172.

Tierney, John. 1984. Paul Erdos is in town. His brain is open. Science (Oct.): 40-47.

Wiles, Andrew. 1995. Modular elliptic curves and Fermat's Last Theorem. Annals of Mathematics (May): 1.

Grossman can be reached at grossman@oakland.edu.

Comments are welcome. Please send messages to Ivars Peterson at ipeterson@maa.org.