If Alice collaborates with Paul Erdős on one paper, and with Bob on another, but Bob never collaborates with Erdős himself, then Bob is given an Erdős number of 2, as he is two steps from Erdős.

* The Erdős Webgraph Server visualizes the distribution of the degrees of the webgraph on the download page.

Paul Erdős has also commented on Ulam's work in mathematics.

* Erdős number, collaborations on mathematical papers with Paul Erdős

* Erdős – Bacon number, the sum of a Bacon number and Erdős number, defined for some who have both collaborated on academic papers and worked in the film industry

Erdős often offered monetary rewards ; the size of the reward depended on the perceived difficulty of the problem.

Furthermore, he proved that as n tends to infinity, the number of primes between 3n and 4n also goes to infinity, thereby generalizing Erdős ' and Ramanujan's results ( see the section on Erdős ' theorems above ).

In combinatorics, the Erdős – Ko – Rado theorem of Paul Erdős, Chao Ko, and Richard Rado is a theorem on intersecting set families.

* The Cameron – Erdős conjecture on sum-free sets of integers, proved by Ben Green.

* The Erdős – Burr conjecture on Ramsey numbers of graphs.

* The Erdős – Faber – Lovász conjecture on coloring unions of cliques.

* The Erdős – Graham conjecture in combinatorial number theory on monochromatic Egyptian fraction representations of unity.

* The Erdős – Gyárfás conjecture on cycles with lengths equal to a power of two in graphs with minimum degree 3.

The Lovász local lemma ( a weaker version was proved in 1975 by László Lovász and Paul Erdős in the article Problems and results on 3-chromatic hypergraphs and some related questions ) allows one to relax the independence condition slightly: As long as the events are " mostly " independent from one another and aren't individually too likely, then there will still be a positive probability that none of them occurs.

Paul Erdős ( 1913 – 1996 ) was an influential and itinerant mathematician, who spent a large portion of his later life living out of a suitcase and writing papers with those of his colleagues willing to give him room and board.

To be assigned an Erdős number, an author must co-write a research paper with an author with a finite Erdős number.

He had 511 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 ( 9267 people as of 2010 ), 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.

He has published more than 200 mathematical papers on these topics together with such notable mathematicians as Béla Bollobás, Stefan Burr, Paul Erdős, Ron Gould, András Gyárfás, Brendan McKay, Cecil Rousseau, Richard Schelp, Miklós Simonovits, Joel Spencer, and Vera Sós.

: Extended to subgraphs: a maximal subgraph disconnected by no less than a k-vertex cut is identical to a maximal subgraph with a minimum number k of vertex-independent paths between any x, y pairs of nodes in t is equivalent to Menger's theorem for finite graphs and is a deep recent result of Ron Aharoni and Eli Berger for infinite graphs ( originally a conjecture proposed by Paul Erdős ):

Graham popularized the concept of the Erdős number, named after the highly prolific Hungarian mathematician Paul Erdős ( 1913 – 1996 ).

Erdős often stayed with Graham, and allowed him to look after his mathematical papers and even his income.

Graham and Erdős visited the young mathematician Jon Folkman when he was hospitalized with brain cancer.

Ronald Graham, his wife Fan Chung, and Paul Erdős, Japan 1986

* Erdős – Graham problem

Fan Chung, her husband Ronald Graham, and Paul Erdős, Japan 1986

The topic of Egyptian fractions has also seen interest in modern number theory ; for instance, the Erdős – Graham conjecture and the Erdős – Straus conjecture concern sums of unit fractions, as does the definition of Ore's harmonic numbers.

A large part of the book concerns Erdős, but a lot of it is about other mathematicians, past and present, including Ronald Graham, Carl Friedrich Gauss, Srinivasa Ramanujan and G. H.

During Folkman's hospitalization, he was visited repeatedly by Ronald Graham and Paul Erdős.

As soon as Folkman received Graham and Erdős at the hospital, Erdős challenged Folkman with mathematical problems, helping to rebuild his confidence.

Szekeres worked closely with many prominent mathematicians throughout his life, including Paul Erdős, Esther Szekeres ( née Esther Klein ), Paul Turán, Béla Bollobás, Ronald Graham, Alf van der Poorten, Miklós Laczkovich, and John Coates.

The well-known aphorism, " A mathematician is a device for turning coffee into theorems ", is probably due to Alfréd Rényi, although it is often attributed to Rényi's colleague Paul Erdős ( and Rényi may have been thinking of Erdős ), who was famous for the many theorems he produced, the number of his collaborations, and his coffee drinking.

Although others before him proved theorems via the probabilistic method ( for example, Szele's 1943 result that there exist tournaments containing a large number of Hamiltonian cycles ), many of the most well known proofs using this method are due to Erdős.

* The Erdős – Lovász conjecture on weak / strong delta-systems (, p. 406 ), proved by Michel Deza.

* March 26-Paul Erdős ( died 1996 ), mathematician.

Most commonly studied is the one proposed by Edgar Gilbert, denoted G ( n, p ), in which every possible edge occurs independently with probability p. A closely related model, the Erdős – Rényi model denoted G ( n, M ), assigns equal probability to all graphs with exactly M edges.

* September 20 – Paul Erdős ( b. 1913 ), mathematician.

Thus, no cardinal can be ω < sub > 1 </ sub >- Erdős in L. While L does contain the initial ordinals of those large cardinals ( when they exist in a supermodel of L ), and they are still initial ordinals in L, it excludes the auxiliary structures ( e. g. measures ) which endow those cardinals with their large cardinal properties.

Highly abundant numbers and several similar classes of numbers were first introduced by Pillai ( 1943 ), and early work on the subject was done by Alaoglu and Erdős ( 1944 ).

Hungar 8 ( 1957 ), 443 – 452 ); this gives him an Erdős number of 1.

Paul Erdős ( in ) observed that, when n is a prime number, the set of n grid points ( i, i < sup > 2 </ sup > mod n ), for 0 ≤ i < n, contains no three collinear points.

The following problem from Paul Erdős is unsolved: Whether for any arbitrarily large N there exists an incongruent covering system the minimum of whose moduli is at least N. It is easy to construct examples where the minimum of the moduli in such a system is 2, or 3 ( Erdős gave an example where the moduli are in the set of the divisors of 120 ; a suitable cover is 0 ( 3 ), 0 ( 4 ), 0 ( 5 ), 1 ( 6 ), 1 ( 8 ), 2 ( 10 ), 11 ( 12 ), 1 ( 15 ), 14 ( 20 ), 5 ( 24 ), 8 ( 30 ), 6 ( 40 ), 58 ( 60 ), 26 ( 120 ) ); D. Swift gave an example where the minimum of the moduli is 4 ( and the moduli are in the set of the divisors of 2880 ).