If E(x) represents Erdős number, then

E(x): H→R

is a mapping, where H is the set of all human beings, and N is the set of nonnegative integers.

H is an increasing variable set with time.

The solution of E(x)=0 has a solution x ∈ K where K is the set of all humans who are not co-authors with Erdős and not any co-author of any co-author of Erdős up to any order!

K is a nonempty set almost all elements in H are in K except a very few.

E(x)=1 has a few numbers of solutions, that is x∈C, where C is the set of co-authors of Erdős and the cardinal number of C is 511.

Now I raise the following question:

Assume that r >1 a positive integer. Is E(x) = r solvable?

In other words Is E(x) onto?

Another important question: What is E(Erdős)?

If the answer is E(Erdős) = 0 because Erdős is not a co-author of himself.

But on the other hand: Erdős is a co-author with some Erdős co-authors, and then E(Erdős) is not zero !

Is it a paradox like that of Russell's paradox?

Best wishes

My hunch is that if you have an Erdös number less than 3 or so, then you may crack jokes based on it, but you need to realize that it's not thanks to you if your number is low. Mine is 4, and I have counted at least 9 routes that gets me there. Trivia, indeed.

I also find it much more enlightening when you pick someone other than Erdös - such as a person that you have connected with at a conference, by chance. Then it can be an eye-opener!

I checked Math Reviews on the issue of single versus co-authorships, and here is the verdict:

"There are about 1.9 million authored items in the Math Reviews database, by a total of about 401,000 different authors. (This includes all books and papers in MR except those items, such as some conference proceedings, that do not have authors.) Approximately 62.4% of these items are by a single author, 27.4% by two authors, 8.0% by three authors, 1.7% by four authors, 0.4% by five authors, and 0.1% by six or more authors. The largest number of authors shown for a single item is in the 20s, but sometimes the author list includes “et al.”, whom we did not count as a real person. The fraction of items authored by just one person has steadily decreased over time, starting out above 90% in the 1940s and currently standing at under 50%."

So, indeed math is still somewhat of a loner subject (with 62.4% of the entire database), but collaborations are getting more and more frequent; in fact it is nowadays more common for two or more to collaborate, while the above in fact also shows that the most frequent type of authorship is *still* the single-author one.

No, you're right - still 2004. We should request an update!

If E(x) represents Erdős number, then

E(x): H→R

is a mapping, where H is the set of all human beings, and N is the set of nonnegative integers.

H is an increasing variable set with time.

The solution of E(x)=0 has a solution x ∈ K where K is the set of all humans who are not co-authors with Erdős and not any co-author of any co-author of Erdős up to any order!

K is a nonempty set almost all elements in H are in K except a very few.

E(x)=1 has a few numbers of solutions, that is x∈C, where C is the set of co-authors of Erdős and the cardinal number of C is 511.

Now I raise the following question:

Assume that r >1 a positive integer. Is E(x) = r solvable?

In other words Is E(x) onto?

Another important question: What is E(Erdős)?

If the answer is E(Erdős) = 0 because Erdős is not a co-author of himself.

But on the other hand: Erdős is a co-author with some Erdős co-authors, and then E(Erdős) is not zero !

Is it a paradox like that of Russell's paradox?

Best wishes

Dear Peter,

Excellent interpretations. This will combine ( Graph theory) with your Z2 action on H.

But first, let me ask : Is H metrizable ? The inventors themselves of (Erdos number) considered E as a (collaborative distance), some type of metric .

Interesting!

Best regards

If you are a mathematician, an Erdos number provides an important signature (characteristic) of that person's work. Being the coauthor of a person who is either the coauthor or is close to (shared some mathematics with) Erdos is indeed significant.

The Erdos number for a mathematician is an indicator of the quality and depth of that person's mathematics relative to the areas of mathematics that were of interest to Erdos himself.

For followers of this thread, a straightforward as well as brief introduction to Erdos numbers is available at

http://mathworld.wolfram.com/ErdosNumber.html

Answering this question needs to consider the other measures, and to see how much coherence exists among these measures.Probably, good Mathematicians in western country specially US & Hungary in which their subjects are near to Combinatorics have good ranks.But I am not so sure for Mathematicians in the other countries or the other fields.

It is interesting to investigate that among French and Russian Mathematicians,

specially in some fields like Geometry, Differential equation or Logic and to compare when the subject is Combinatorics or Computer Science.

I like your comparison, I once slept in the house of Andrew Granville in the bed that the week before Erdos slept in...this was in 1994 I think, when Andrew was prof in Athens, Georgia

What kind of topological properties it may have? Can you please discuss some of those.

Tapas Chatterjee , have a look at this article on RG:

Article Generalized Erdős numbers for network analysis

https://or-zse.hu/hacofe/vol10/erdospal2015.htm

Did conduct biographical and professional research on Erdös,

with the Jewish University Journal (Budapest). The Erdös number was misinterpreted by managerial circles as networking capacity, with repect to quantifying communication in a network of collaborators. This misunderstanding fully undermines the original approach of Erdös, saying: my mind or my door is open. So, for Erdös, it was essential to share the hotel room for the sake of scientific communication; for Erdös, a mathematician was a living organism that was capable to convert drinking coffee into algorithms.Consequently, you would have been invited into the hotel room by Erdös, with no chance to fall asleep.