Rational Erd} Os Numbers

The concept of Erd} os number is well known among mathematicians and somewhat known more generally (see, for example, de Castro & Grossman, 1999] or the web site Paul Erd} os, who died in late 1996, was a mathematician who was author or coauthor|most often the latter|of nearly 1500 papers and had nearly 500 coauthors. By deenition, Erd} os had Erd} os number zero, his collaborators had Erd} os number one, a person who not collaborated with Erd} os, but had collaborated with one of his collaborators has Erd} os number two and so on. For example, I have written a joint paper with Michael Makkai, who has a joint paper with Erd} os and so Makkai has Erd} os number one and mine is two. But there is something not entirely rational about this assignment of numbers. Surely a person who has written, say, ve joint papers with Erd} os is more closely associated with Erd} os than someone who has written only one and therefore ought to have a smaller Erd} os number. In fact, it seems entirely reasonable that such a person be assigned the Erd} os number 1=5. If a person has written two joint papers with someone whose Erd} os number is 1, then his Erd} os number ought to be 3/2. If you have written two joint papers with someone who has written three joint papers with Erd} os, then your Erd} os number should be 5/6 and so on. But what if you have written two joint papers with one Erd} os coauthor and three with someone whose Erd} os number is 2? This leads to an obvious iteration procedure for calculating these new,