Analyzing the Erdös Star Cluster

Just as actors have Kevin Bacon, mathematicians have Paul Erdös. People in each field love to calculate their "degree of separation" from co-stars or co-authors. Now two Erdös fans have shown that all the mathematicians who have won the prestigious Fields Medal have a collaboration trail that leads to Erdös. And the web extends to a surprising number of Nobel laureates in everything from biochemistry to linguistics.

What's your Erdös number? If you've co-authored a paper with the prolific Hungarian mathematician, you get a 1; if you've collaborated with an Erdös-1 mathematician, you've earned a 2; collaboration with a 2 gives you a 3, and so on. Since Erdös tended to publish with top-flight talent, the lower your number, the more closely you're associated with an elite group of extremely important mathematicians. At last count, the Erdös-1 club had only 507 members. That makes a low Erdös number a matter of pride--so much so that counterfeit collaborations may have been concocted to lower someone's number, says University of Chicago mathematician Lázsló Babai. He's seen at least one paper with Erdös's name as co-author that he found "a bit fishy."

Mathematicians, no less proud than the rest of us, are fond of analyzing Erdös numbers. The latest effort, appearing in the December issue of the journal of the Columbian Academy of Sciences, shows that all 42 mathematicians who have won the Fields medal--mathematics' highest honor--have some connection to Erdös, and usually a close one. (In math talk, they have a finite Erdös number, which tends to be small.) "All have a five or less, and only one or two have a five," says co-author Jerrold Grossman of Oakland University in Michigan. More than 60 Nobel laureates, too, have finite Erdös numbers, including many in fields far removed from mathematics. "For example, Watson and Crick have Erdös numbers of seven and eight," says Grossman. "There's a lot of math cross-over between different fields."

Erdös died in 1996 after having published close to 1500 papers. But his nonstop travels left a legacy of unpublished work--so his name keeps appearing on papers to this day. According to Grossman, 25 new Erdös papers entered his database this year, promoting 15 mathematicians to the exalted Erdös-1 status. Babai believes these papers are genuine. Probably. "I could imagine two or three fakes slip through," he says.

Posted in Math