Editing Erdős number (section)

==Definition and application in mathematics==
[[File:Erdosnumber-mlng.svg|thumb|If [[Alice and Bob|Alice]] collaborates with Paul Erdős on one paper, and with Bob on another, but Bob never collaborates with Erdős directly, then Alice is given an Erdős number of 1 and Bob is given an Erdős number of 2, as he is two steps from Erdős.]]

To be assigned an Erdős number, someone must be a coauthor of a research paper with another person who has a finite Erdős number. Paul Erdős himself is assigned an Erdős number of zero. A certain author's Erdős number is one greater than the lowest Erdős number of any of their collaborators; for example, an author who has coauthored a publication with Erdős would have an Erdős number of 1. The [[American Mathematical Society]] provides a free online tool to determine the collaboration distance between two mathematical authors listed in the ''[[Mathematical Reviews]]'' catalogue.<ref name="Collaboration distance">{{cite web|url=https://www.ams.org/mathscinet/collaborationDistance.html|title= Collaboration Distance|work=[[MathSciNet]]|publisher=American Mathematical Society}}</ref>

Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 509 direct collaborators;<ref name="Erdős Number Project"/> these are the people with Erdős number 1. The people who have collaborated with them (but not with Erdős himself) have an Erdős number of 2 (12,600 people as of 7 August 2020<ref name="Erdős Number Project File Erdos2">[https://www.oakland.edu/enp/thedata/erdos2/ Erdos2], Version 2020, 7 August 2020.</ref>), those who have collaborated with people who have an Erdős number of 2 (but not with Erdős or anyone with an Erdős number of 1) have an Erdős number of 3, and so forth. A person with no such coauthorship chain connecting to Erdős has an Erdős number of [[infinity]] (or an [[defined and undefined|undefined]] one). Since the death of Paul Erdős, the lowest Erdős number that a new researcher can obtain is 2.

There is room for ambiguity over what constitutes a link between two authors. The American Mathematical Society collaboration distance calculator uses data from ''Mathematical Reviews'', which includes most mathematics journals but covers other subjects only in a limited way, and which also includes some non-research publications. The Erdős Number Project web site says:{{blockquote|... One drawback of the MR system is that it considers all jointly authored works as providing legitimate links, even articles such as obituaries, which are not really joint research. ...<ref name="Oakland University Erdös Number compute">{{cite web | title=Compute your Erdös number - The Erdös Number Project | website=Oakland University | date=1999-02-22 | url=https://www.oakland.edu/enp/compute/ | access-date=2022-10-15}}</ref>}} It also says:

{{Blockquote|... Our criterion for inclusion of an edge between vertices u and v is some research collaboration between them resulting in a published work. Any number of additional co-authors is permitted,...}}
but excludes non-research publications such as elementary textbooks, joint editorships, obituaries, and the like. The "Erdős number of the second kind" restricts assignment of Erdős numbers to papers with only two collaborators.<ref>Grossman ''et al.''  "[http://www.oakland.edu/?id=9569&sid=243#en2k Erdős numbers of the second kind]," in ''Facts about Erdős Numbers and the Collaboration Graph''. [http://www.oakland.edu/enp The Erdős Number Project], [[Oakland University]], USA. Retrieved July 25, 2009.</ref>

The Erdős number was most likely first defined in print by Casper Goffman, an [[Mathematical analysis|analyst]] whose own Erdős number is 2.<ref name="Erdős Number Project File Erdos2"/> Goffman published his observations about Erdős' prolific collaboration in a 1969 article entitled "''And what is your Erdős number?''"<ref>{{cite journal|last=Goffman|first=Casper|title=And what is your Erdős number? |journal=[[The American Mathematical Monthly]] |volume=76 |year=1969 |doi=10.2307/2317868|page=791|jstor=2317868|issue=7}}</ref> See also some comments in an obituary by Michael Golomb.<ref>{{Cite web|url=https://www.math.purdue.edu/about/purview/fall96/paul-erdos.html|title=Paul Erdös at Purdue|website=www.math.purdue.edu}}</ref>

The median Erdős number among Fields Medalists is as low as 3.<ref name="paths"/> Fields Medalists with Erdős number 2 include [[Atle Selberg]], [[Kunihiko Kodaira]], [[Klaus Roth]], [[Alan Baker (mathematician)|Alan Baker]], [[Enrico Bombieri]], [[David Mumford]], [[Charles Fefferman]], [[William Thurston]], [[Shing-Tung Yau]], [[Jean Bourgain]], [[Richard Borcherds]], [[Manjul Bhargava]], [[Jean-Pierre Serre]] and [[Terence Tao]]. There are no Fields Medalists with Erdős number 1;<ref name="project">{{cite web|url=http://www.oakland.edu/enp/erdpaths/|title=Paths to Erdös|work=The Erdös Number Project|publisher=Oakland University}}</ref> however, [[Endre Szemerédi]] is an [[Abel Prize]] Laureate with Erdős number 1.<ref name="trails"/>