The Erdős number (Hungarian: [ˈɛrdøːʃ]) describes the "collaborative distance" between mathematician Paul Erdős and another person, as measured by authorship of mathematical papers. The same principle has been applied in other fields where a particular individual has collaborated with a large and broad number of peers.

Paul Erdős (1913–1996) was an influential Hungarian mathematician who, in the latter part of his life, spent a great deal of time writing papers with a large number of colleagues—more than 500—working on solutions to outstanding mathematical problems. He published more papers during his lifetime (at least 1,525) than any other mathematician in history. (Leonhard Euler published more total pages of mathematics but fewer separate papers: about 800.) Erdős spent most of his career with no permanent home or job. He traveled with everything he owned in two suitcases, and would visit mathematicians with whom he wanted to collaborate, often unexpectedly, and expect to stay with them.

The idea of the Erdős number was originally created by the mathematician's friends as a tribute to his enormous output. Later it gained prominence as a tool to study how mathematicians cooperate to find answers to unsolved problems. Several projects are devoted to studying connectivity among researchers, using the Erdős number as a proxy. For example, Erdős collaboration graphs can tell us how authors cluster, how the number of co-authors per paper evolves over time, or how new theories propagate.

Several studies have shown that leading mathematicians tend to have particularly low Erdős numbers. The median Erdős number of Fields Medalists is 3. Only 7,097 (about 5% of mathematicians with a collaboration path) have an Erdős number of 2 or lower. As time passes, the lowest Erdős number that can still be achieved will necessarily increase, as mathematicians with low Erdős numbers die and become unavailable for collaboration. Still, historical figures can have low Erdős numbers. For example, renowned Indian mathematician Srinivasa Ramanujan has an Erdős number of only 3 (through G. H. Hardy, Erdős number 2), even though Paul Erdős was only 7 years old when Ramanujan died.

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.

Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 509 direct collaborators; 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), 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 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:

... 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. ...