“爱多士数”的版本间的差异

2020年10月10日 (六) 15:25的版本

此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。

保罗·埃尔德什Paul Erdős摄于1992年

The Erdős number (模板:IPA-hu) describes the "collaborative distance" between mathematician 模板:Nobr 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.

The Erdős number () describes the "collaborative distance" between mathematician 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.

埃尔德什数Erdős number（匈牙利语：[ˈɛrdøːʃ]）描述了数学家保罗·埃尔德什Paul Erdős与另一个人之间的“协作距离”，这是根据数学论文的著作权来衡量的。该原则应用于很多其他领域，意指特定某个人与众多同行之间的合作。

Overview

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, 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 a large portion of his later life living out of a suitcase, visiting his over 500 collaborators around the world.

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, working on solutions to outstanding mathematical problems. He published more papers during his lifetime (at least 1,525) than any other mathematician in history. Erdős spent a large portion of his later life living out of a suitcase, visiting his over 500 collaborators around the world.

保罗·埃尔德什（Paul Erdős，1913年至1996年）是一位具有很大影响力的匈牙利数学家，他花费了一生中大量的时间与很多同事撰写论文，致力于解决困扰已久的疑难数学问题。他一生中发表的论文（至少1,525件）比历史上其他任何数学家都多。（莱昂哈德·欧拉（Leonhard Euler）发表过更多的数学论文，但单独的论文却较少：大约800篇。），而埃尔德什（Erdős）的大部分时间都生活在手提箱里，他拜访过全球500多个合作者。

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.

埃尔德什数的概念最初是由数学家的朋友们提出来的，以赞扬保罗·埃尔德什的巨大成就。后来，它演变为研究数学家如何通过合作来解决问题的的工具而受到重视。有几个项目专门通过使用埃尔德什数作为代理来研究人员之间的连通性。例如，埃尔德什合作图可以告诉我们作者是如何聚集在一起的，每篇论文的共同作者数量随时间变化或新理论的产生又是如何传播的。

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 smallest 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.

多项研究表明，领先的数学家往往具有极低的埃尔德什数。费尔兹奖Fields Medalists的埃尔德什中位数是3。只有7,097名（拥有合作经历的数学家中约5％）的埃尔德什数为2或更低。随着时间的流逝，低埃尔德什数的数学家因死亡而无法进行协作，最小埃尔德什数（仍然存在）必然会增加。即使历史人物仍可能一直具有较低的埃尔德什数。例如，印度著名数学家Srinivasa Ramanujan的埃尔德什数仅为3（通过G. H. Hardy，Erdős数为2），尽管Ramanujan去世时保罗·埃尔德什只有7岁。

Definition and application in mathematics 数学的定义与应用

如果爱丽丝在一张纸上与保罗·埃尔德什合作，在另一张纸上与鲍勃合作，但是鲍勃从未与埃尔德什本人合作，那么爱丽丝的埃尔德什数为1，而鲍勃的埃尔德什数为2，因为他离埃尔德什有两步。

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 has an Erdős number of zero. Anybody else's Erdős number is $k + 1$ where $k$ is the lowest Erdős number of any coauthor. The American Mathematical Society provides a free online tool to determine the Erdős number of every mathematical author listed in the Mathematical Reviews catalogue.

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 has an Erdős number of zero. Anybody else's Erdős number is where is the lowest Erdős number of any coauthor. The American Mathematical Society provides a free online tool to determine the Erdős number of every mathematical author listed in the Mathematical Reviews catalogue.

要分配一个埃尔德什数，某人必须与另一个具有有限埃尔德什数的人共同撰写研究论文。保罗·埃尔德什的埃尔德什数为零。其他人的埃尔德什数为k+1，其中k是任何合著者中最低的埃尔德什数。美国数学学会提供免费的在线工具，可确定《数学评论》目录中列出的每个数学作者的埃尔德什数。

Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. 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 (11,009 people as of 2015), 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.

Erdős wrote around 1,500 mathematical articles in his lifetime, mostly co-written. He had 511 direct collaborators;), 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.

埃尔德什一生撰写了约1500篇数学文章，其中大部分是合作的。他有511个直接合作者；这些是埃尔德什数为1的人。与这些人合作（但未与埃尔德什本人合作）的人所拥有的埃尔德什数为2（截至2020年8月7日为12,600人），而与埃尔德什数为2的人合作的人（但与埃尔德什或埃尔德什数为1的任何人无合作关系），其埃尔德什数为3，依此类推。没有此类共同作者链接能指向埃尔德什的人，其埃尔德什数为无穷大（或未定义）。自保罗·埃尔德什逝世以来，新研究员可获得的最低Erdős数为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^{[citation needed]}. The Erdős Number Project web site says:

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:

关于具体由什么构成两位作者之间的联系，众说纷纭。美国数学学会的“协作距离计算器”使用的是来自《数学评论》的数据，包括大多数数学期刊，但仅以有限的方式涵盖了其他主题，同时还包括一些非研究出版物。埃尔德什数项目官方网站Erdős Number Project表示：

... 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,...

...我们在顶点u和v之间共有的包含边标准是，它们之间的某些研究合作导致了发表的作品。允许任何数量的其他共同作者，...

but they do not include 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.

但它们不包括非研究性出版物，例如教科书，联合编辑，讣告等。“第二种埃尔德什数”将其分配给只有两个合作者的论文。

The Erdős number was most likely first defined in print by Casper Goffman, an analyst whose own Erdős number is 2. Goffman published his observations about Erdős' prolific collaboration in a 1969 article entitled "And what is your Erdős number?" See also some comments in an obituary by Michael Golomb.

埃尔德什数很可能最早由卡斯珀·高夫曼Casper Goffman定义，他自己的埃尔德什数为2。高夫曼在1969年发表的一篇文章中表示了他对埃尔德什多产合作的看法，“您的埃尔德什数是多少？”另请参阅迈克尔·哥伦布Michael Golomb在讣告中的一些评论。

The median Erdős number among Fields medalists is as low as 3. Fields medalists with Erdős number 2 include Atle Selberg, Kunihiko Kodaira, Klaus Roth, 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; however, Endre Szemerédi is an Abel Prize Laureate with Erdős number 1.

The median Erdős number among Fields medalists is as low as 3.Fields medalists with Erdős number 2 include Atle Selberg, Kunihiko Kodaira, Klaus Roth, 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; however, Endre Szemerédi is an Abel Prize Laureate with Erdős number 1.

Fields奖牌获得者的埃尔德什中位数低至3。埃尔德什排名第二的奖牌获得者包括Atle Selberg，Kunihiko Kodaira，Klaus Roth，Alan Baker，Enrico Bombieri，David Mumford，Charles Fefferman，William Thurston，Shing-Tung Tung，Jean Bourgain，Richard Borcherds，Manjul Bhargava，Jean-Pierre Serre和陶哲轩。Fields的获得者中没有人的Erdős为1。但是，恩德雷·塞梅雷迪（Endre Szemerédi）是阿贝尔奖获得者，其埃尔德什数为1。

Most frequent Erdős collaborators 最频繁的埃尔德什合作者

While Erdős collaborated with hundreds of co-authors, there were some individuals with whom he co-authored dozens of papers. This is a list of the ten persons who most frequently co-authored with Erdős and their number of papers co-authored with Erdős (i.e. their number of collaborations).

虽然埃尔德什与数百位合著者合作，但其中一些人与他合作过数十篇论文。以下是最经常与埃尔德什合作的十人列表，以及与埃尔德什合作的论文数量（即合作数量）。

{ | class = “ wikitable sortable”

Co-author	Number of collaborations	Co-author	Number of collaborations
András Sárközy	62	András Sárközy	62	András Sárközy	62
András Hajnal	56	András Hajnal	56	András Hajnal	56
Ralph Faudree	50	Ralph Faudree	50
Richard Schelp	42	Richard Schelp	42	\| 42
Cecil C. Rousseau	35	Cecil C. Rousseau	35
Vera T. Sós	35	Vera T. Sós	35	Vera T. Sós	35
Alfréd Rényi	32	Alfréd Rényi	32	Alfréd Rényi	32
Pál Turán	30	Pál Turán	30	Pál Turán	30
Endre Szemerédi	29	Endre Szemerédi	29	Endre Szemerédi	29
Ronald Graham	28	Ronald Graham	28

|}

Related fields

模板:As of, all Fields Medalists have a finite Erdős number, with values that range between 2 and 6, and a median of 3. In contrast, the median Erdős number across all mathematicians (with a finite Erdős number) is 5, with an extreme value of 13.^[1] The table below summarizes the Erdős number statistics for Nobel prize laureates in Physics, Chemistry, Medicine and Economics.^[2] The first column counts the number of laureates. The second column counts the number of winners with a finite Erdős number. The third column is the percentage of winners with a finite Erdős number. The remaining columns report the minimum, maximum, average and median Erdős numbers among those laureates.

, all Fields Medalists have a finite Erdős number, with values that range between 2 and 6, and a median of 3. In contrast, the median Erdős number across all mathematicians (with a finite Erdős number) is 5, with an extreme value of 13. The table below summarizes the Erdős number statistics for Nobel prize laureates in Physics, Chemistry, Medicine and Economics. The first column counts the number of laureates. The second column counts the number of winners with a finite Erdős number. The third column is the percentage of winners with a finite Erdős number. The remaining columns report the minimum, maximum, average and median Erdős numbers among those laureates.

所有田径奖获得者都有一个有限的 erd s 数，数值范围在2到6之间，中位数为3。相比之下，所有数学家的 erd 数的中位数(有限 erd 数)是5，极值是13。下表总结了 erd 的数字统计诺贝尔奖获得者在物理，化学，医学和经济学。第一栏计算获奖者的数量。第二列用有限的 erd 数计算赢家的数目。第三列是有限 erd 数目的获胜者的百分比。剩下的柱子报告的最低，最高，平均和中位数 erd 的数字在这些获奖者。

{ | class = “ wikitable sortable”

Statistics on Mathematical Collaboration, 1903-2016
+ 数学合作统计，1903-2016				#Laureates	#Laureates	* 获奖者	#Erdős	#Erdős	#Erdős	%Erdős	%Erdős	%Erdős	Min	Min	最小	Max	Max	麦克斯	Average	Average	平均数	Median	Median	中位数
Fields Medal	Fields Medal 菲尔兹奖	56	56	56	56	56	56	100.0%	100.0%	100.0%	2	2	2	6	6	6	3.36	3.36	3.36	3	3	3
Nobel Economics	Nobel Economics 诺贝尔经济学奖	76	76	76	47	47	47	61.84%	61.84%	61.84%	2	2	2	8	8	8	4.11	4.11	4.11	4	4	4
Nobel Chemistry	Nobel Chemistry	诺贝尔化学奖	172	172	172	42	42	42	24.42%	24.42%	24.42%	3	3	3	10	10	10	5.48	5.48	5.48	5	5	5
Nobel Medicine	Nobel Medicine 诺贝尔医学奖	210	210	210	58	58	58	27.62%	27.62%	27.62%	3	3	3	12	12	12	5.50	5.50	5.50	5	5	5
Nobel Physics	Nobel Physics 诺贝尔物理学奖	200	200	200	159	159	159	79.50%	79.50%	79.50%	2	2	2	12	12	12	5.63	5.63	5.63	5	5	5

|}

Physics

Among the Nobel Prize laureates in Physics, Albert Einstein and Sheldon Lee Glashow have an Erdős number of 2. Nobel Laureates with an Erdős number of 3 include Enrico Fermi, Otto Stern, Wolfgang Pauli, Max Born, Willis E. Lamb, Eugene Wigner, Richard P. Feynman, Hans A. Bethe, Murray Gell-Mann, Abdus Salam, Steven Weinberg, Norman F. Ramsey, Frank Wilczek, and David Wineland. Fields Medal-winning physicist Ed Witten has an Erdős number of 3.^[3]

Among the Nobel Prize laureates in Physics, Albert Einstein and Sheldon Lee Glashow have an Erdős number of 2. Nobel Laureates with an Erdős number of 3 include Enrico Fermi, Otto Stern, Wolfgang Pauli, Max Born, Willis E. Lamb, Eugene Wigner, Richard P. Feynman, Hans A. Bethe, Murray Gell-Mann, Abdus Salam, Steven Weinberg, Norman F. Ramsey, Frank Wilczek, and David Wineland. Fields Medal-winning physicist Ed Witten has an Erdős number of 3.

在诺贝尔物理学奖获得者中，阿尔伯特 · 爱因斯坦和谢尔登·格拉肖的 erd 数是2。诺贝尔奖获得者包括 Enrico Fermi，Otto Stern，Wolfgang Pauli，Max Born，Willis e. Lamb，Eugene Wigner，理查德·费曼，Hans a. Bethe，默里·盖尔曼，Abdus Salam，Steven Weinberg，Norman f. Ramsey，Frank Wilczek，and David Wineland。菲尔兹奖获得者，物理学家爱德 · 威滕的爱德华数为3。

Biology

Computational biologist Lior Pachter has an Erdős number of 2.^[4] Evolutionary biologist Richard Lenski has an Erdős number of 3, having co-authored a publication with Lior Pachter and with mathematician Bernd Sturmfels, each of whom has an Erdős number of 2.^[5]

Computational biologist Lior Pachter has an Erdős number of 2. Evolutionary biologist Richard Lenski has an Erdős number of 3, having co-authored a publication with Lior Pachter and with mathematician Bernd Sturmfels, each of whom has an Erdős number of 2.

计算生物学家 Lior Pachter 的 erd 数为2。进化生物学家理查德 · 伦斯基的爱因斯坦数为3，他曾与莱尔 · 帕切特和数学家贝恩德 · 斯图尔斯合著了一本出版物，每本书的爱因斯坦数为2。

Finance and economics

There are at least two winners of the Nobel Prize in Economics with an Erdős number of 2: Harry M. Markowitz (1990) and Leonid Kantorovich (1975). Other financial mathematicians with Erdős number of 2 include David Donoho, Marc Yor, Henry McKean, Daniel Stroock, and Joseph Keller.

至少有两位诺贝尔经济学奖获得者的 erd 数量是2: Harry m. Markowitz (1990)和列昂尼德·坎托罗维奇 · 马科维茨(1975)。其他拥有 erd 数2的金融数学家包括 David Donoho，Marc Yor，Henry McKean，Daniel Stroock 和 Joseph Keller。

Nobel Prize laureates in Economics with an Erdős number of 3 include Kenneth J. Arrow (1972), Milton Friedman (1976), Herbert A. Simon (1978), Gerard Debreu (1983), John Forbes Nash, Jr. (1994), James Mirrlees (1996), Daniel McFadden (1996), Daniel Kahneman (2002), Robert J. Aumann (2005), Leonid Hurwicz (2007), Roger Myerson (2007), Alvin E. Roth (2012), and Lloyd S. Shapley (2012) and Jean Tirole (2014).^[6]

Nobel Prize laureates in Economics with an Erdős number of 3 include Kenneth J. Arrow (1972), Milton Friedman (1976), Herbert A. Simon (1978), Gerard Debreu (1983), John Forbes Nash, Jr. (1994), James Mirrlees (1996), Daniel McFadden (1996), Daniel Kahneman (2002), Robert J. Aumann (2005), Leonid Hurwicz (2007), Roger Myerson (2007), Alvin E. Roth (2012), and Lloyd S. Shapley (2012) and Jean Tirole (2014).

诺贝尔经济学奖获得者包括肯尼斯·约瑟夫·阿罗(1972) ，米尔顿弗里德曼(1976) ，赫伯特·西蒙(1978) ，Gerard Debreu (1983) ，约翰·福布斯·纳什，jr. 。(1994)、 James Mirrlees (1996)、 Daniel McFadden (1996)、 Daniel Kahneman (2002)、 Robert j. Aumann (2005)、,leonid Hurwicz (2007) ，Roger Myerson (2007) ，Alvin e. Roth (2012) ，Lloyd s. Shapley (2012)和 Jean Tirole (2014)。

Some investment firms have been founded by mathematicians with low Erdős numbers, among them James B. Ax of Axcom Technologies, and James H. Simons of Renaissance Technologies, both with an Erdős number of 3.^[7]^[8]

Some investment firms have been founded by mathematicians with low Erdős numbers, among them James B. Ax of Axcom Technologies, and James H. Simons of Renaissance Technologies, both with an Erdős number of 3.

有些投资公司是由数量较少的数学家创立的，其中包括詹姆斯 · b。的 Ax 和文艺复兴科技的 James h. Simons，他们的 erd 数都是3。

Philosophy

Since the more formal versions of philosophy share reasoning with the basics of mathematics, these fields overlap considerably, and Erdős numbers are available for many philosophers.^[9] Philosopher John P. Burgess has an Erdős number of 2.^[4] Jon Barwise and Joel David Hamkins, both with Erdős number 2, have also contributed extensively to philosophy, but are primarily described as mathematicians.

Since the more formal versions of philosophy share reasoning with the basics of mathematics, these fields overlap considerably, and Erdős numbers are available for many philosophers. Philosopher John P. Burgess has an Erdős number of 2. Jon Barwise and Joel David Hamkins, both with Erdős number 2, have also contributed extensively to philosophy, but are primarily described as mathematicians.

由于更正式的哲学版本与数学基础共享推理，这些领域有相当大的重叠，而 erd 的数字对许多哲学家来说是可用的。哲学家约翰 · p · 伯吉斯的 erd 数为2。乔恩 · 巴韦斯和乔尔 · 大卫 · 汉姆金斯，都是厄尔德二号，也对哲学做出了广泛的贡献，但主要被描述为数学家。

Law

Judge Richard Posner, having coauthored with Alvin E. Roth, has an Erdős number of at most 4. Roberto Mangabeira Unger, a politician, philosopher and legal theorist who teaches at Harvard Law School, has an Erdős number of at most 4, having coauthored with Lee Smolin.

法官理查德 · 波斯纳与阿尔文 · e · 罗斯合著了本书，本书的爱尔兰数量最多不超过4本。是一位政治家、哲学家和法律理论家，在哈佛大学法学院任教，他与 Lee Smolin 合著的《昂格尔多达4本。

Politics

Angela Merkel, Chancellor of Germany from 2005 to the present, has an Erdős number of at most 5.^[10]

Angela Merkel, Chancellor of Germany from 2005 to the present, has an Erdős number of at most 5.

从2005年到现在的德国总理，她的 erd 数字最多不超过5。

Engineering

Some fields of engineering, in particular communication theory and cryptography, make direct use of the discrete mathematics championed by Erdős. It is therefore not surprising that practitioners in these fields have low Erdős numbers. For example, Robert McEliece, a professor of electrical engineering at Caltech, had an Erdős number of 1, having collaborated with Erdős himself.^[11] Cryptographers Ron Rivest, Adi Shamir, and Leonard Adleman, inventors of the RSA cryptosystem, all have Erdős number 2.^[4]

Some fields of engineering, in particular communication theory and cryptography, make direct use of the discrete mathematics championed by Erdős. It is therefore not surprising that practitioners in these fields have low Erdős numbers. For example, Robert McEliece, a professor of electrical engineering at Caltech, had an Erdős number of 1, having collaborated with Erdős himself. Cryptographers Ron Rivest, Adi Shamir, and Leonard Adleman, inventors of the RSA cryptosystem, all have Erdős number 2.

一些工程领域，特别是通信理论和密码学，直接利用 erd 所拥护的离散数学。因此，这些领域的从业人员的 erd 值偏低就不足为奇了。例如，加州理工学院的电气工程教授罗伯特 · 麦克里斯与爱尔德本人合作，得到了爱尔德数1。RSA 密码系统的发明者——密码学家罗恩 · 里维斯特、阿迪 · 沙米尔和伦纳德 · 阿德曼都有 erd 数2。

Social network analysis

Anthropologist Douglas R. White has an Erdős number of 2 via graph theorist Frank Harary.^[12]^[13] Sociologist Barry Wellman has an Erdős number of 3 via social network analyst and statistician Ove Frank,^[14] another collaborator of Harary's.^[15]

Anthropologist Douglas R. White has an Erdős number of 2 via graph theorist Frank Harary. Sociologist Barry Wellman has an Erdős number of 3 via social network analyst and statistician Ove Frank, another collaborator of Harary's.

人类学家道格拉斯 · r · 怀特通过图论家弗兰克 · 哈拉里给出了 erd 数2。社会学家巴里 · 韦尔曼通过社交网络分析师和统计学家奥夫 · 弗兰克得到了 erd 数字3，奥夫 · 弗兰克是哈拉里的另一个合作者。

Linguistics

The Romanian mathematician and computational linguist Solomon Marcus had an Erdős number of 1 for a paper in Acta Mathematica Hungarica that he co-authored with Erdős in 1957.^[16]

The Romanian mathematician and computational linguist Solomon Marcus had an Erdős number of 1 for a paper in Acta Mathematica Hungarica that he co-authored with Erdős in 1957.

罗马尼亚数学家、计算语言学家所罗门 · 马库斯在1957年与厄尔德合著的《数学学报》上的一篇论文中，给出了厄尔德数1。

Impact

Paul Erdős in 1985 at the University of Adelaide teaching Terence Tao, who was then 10 years old. Tao became a math professor at UCLA, received the Fields Medal in 2006, and was elected a Fellow of the Royal Society in 2007. His Erdős number is 2

1985年，Paul erd 在阿德莱德大学教导10岁的 Terence Tao。陶成为加州大学洛杉矶分校的数学教授，2006年获得菲尔兹奖，2007年被选为皇家学会会员。他的 erd 数字是2]

Erdős numbers have been a part of the folklore of mathematicians throughout the world for many years. Among all working mathematicians at the turn of the millennium who have a finite Erdős number, the numbers range up to 15, the median is 5, and the mean is 4.65;^[17] almost everyone with a finite Erdős number has a number less than 8. Due to the very high frequency of interdisciplinary collaboration in science today, very large numbers of non-mathematicians in many other fields of science also have finite Erdős numbers.^[18] For example, political scientist Steven Brams has an Erdős number of 2. In biomedical research, it is common for statisticians to be among the authors of publications, and many statisticians can be linked to Erdős via John Tukey, who has an Erdős number of 2. Similarly, the prominent geneticist Eric Lander and the mathematician Daniel Kleitman have collaborated on papers,^[19]^[20] and since Kleitman has an Erdős number of 1,引用错误：没有找到与</ref>对应的<ref>标签 a large fraction of the genetics and genomics community can be linked via Lander and his numerous collaborators. Similarly, collaboration with Gustavus Simmons opened the door for

</ref> a large fraction of the genetics and genomics community can be linked via Lander and his numerous collaborators. Similarly, collaboration with Gustavus Simmons opened the door for

大部分的遗传学和基因组学社区可以通过 Lander 和他的众多合作者联系起来。同样的，与古斯塔夫斯·希门斯的合作也打开了一扇大门

Erdős numbers within the cryptographic research community, and many linguists have finite Erdős numbers, many due to chains of collaboration with such notable scholars as Noam Chomsky (Erdős number 4),^[21] William Labov (3),^[22] Mark Liberman (3),^[23] Geoffrey Pullum (3),^[24] or Ivan Sag (4).^[25] There are also connections with arts fields.^[26]

Erdős numbers within the cryptographic research community, and many linguists have finite Erdős numbers, many due to chains of collaboration with such notable scholars as Noam Chomsky (Erdős number 4), William Labov (3), Mark Liberman (3), Geoffrey Pullum (3), or Ivan Sag (4). There are also connections with arts fields.

Erd 的数字在密码学研究领域，许多语言学家都有有限的 erd 数字，许多是由于与著名学者 Noam Chomsky (erd 数字4) ，William Labov (3) ，Mark Liberman (3) ，Geoffrey Pullum (3) ，或 Ivan Sag (4)的合作链。与艺术领域也有联系。

According to Alex Lopez-Ortiz, all the Fields and Nevanlinna prize winners during the three cycles in 1986 to 1994 have Erdős numbers of at most 9.

根据 Alex Lopez-Ortiz 的说法，在1986年到1994年的3个赛季中，所有的费尔德斯和奈望林纳奖冠军的 erd 数量最多不超过9。

Earlier mathematicians published fewer papers than modern ones, and more rarely published jointly written papers. The earliest person known to have a finite Erdős number is either Antoine Lavoisier (born 1743, Erdős number 13), Richard Dedekind (born 1831, Erdős number 7), or Ferdinand Georg Frobenius (born 1849, Erdős number 3), depending on the standard of publication eligibility.^[27]

Earlier mathematicians published fewer papers than modern ones, and more rarely published jointly written papers. The earliest person known to have a finite Erdős number is either Antoine Lavoisier (born 1743, Erdős number 13), Richard Dedekind (born 1831, Erdős number 7), or Ferdinand Georg Frobenius (born 1849, Erdős number 3), depending on the standard of publication eligibility.

早期数学家发表的论文比现代数学家少，联合发表的论文也更少。根据出版资格标准的不同，已知最早拥有有限 erd 数目的人要么是安托万-洛朗·德·拉瓦锡(出生于1743年，erd 数目13) ，要么是理查德·戴德金(出生于1831年，erd 数目7) ，要么是费迪南德·格奥尔格·弗罗贝尼乌斯(出生于1849年，erd 数目3)。

Martin Tompa^[28] proposed a directed graph version of the Erdős number problem, by orienting edges of the collaboration graph from the alphabetically earlier author to the alphabetically later author and defining the monotone Erdős number of an author to be the length of a longest path from Erdős to the author in this directed graph. He finds a path of this type of length 12.

Martin Tompa proposed a directed graph version of the Erdős number problem, by orienting edges of the collaboration graph from the alphabetically earlier author to the alphabetically later author and defining the monotone Erdős number of an author to be the length of a longest path from Erdős to the author in this directed graph. He finds a path of this type of length 12.

Martin Tompa 提出了 erd 数问题的一个有向图版本，通过定向协作图的边，从字母顺序的前作者到字母顺序的后作者，并定义单调的作者 erd 数为从 Erdős 到作者的最长路径的长度。他找到了一条长度为12的路径。

Also, Michael Barr suggests "rational Erdős numbers", generalizing the idea that a person who has written p joint papers with Erdős should be assigned Erdős number 1/p. From the collaboration multigraph of the second kind (although he also has a way to deal with the case of the first kind)—with one edge between two mathematicians for each joint paper they have produced—form an electrical network with a one-ohm resistor on each edge. The total resistance between two nodes tells how "close" these two nodes are.

Also, Michael Barr suggests "rational Erdős numbers", generalizing the idea that a person who has written p joint papers with Erdős should be assigned Erdős number 1/p. From the collaboration multigraph of the second kind (although he also has a way to deal with the case of the first kind)—with one edge between two mathematicians for each joint paper they have produced—form an electrical network with a one-ohm resistor on each edge. The total resistance between two nodes tells how "close" these two nodes are.

此外，迈克尔 · 巴尔还提出了“合理的 erd 数字” ，概括了这样一个观点，即一个与 erd s 共同撰写了 p 篇论文的人应该被赋予 erd s 1/p。从第二种合作多重图(虽然他也有办法处理第一种情况)ーー两个数学家为他们生产的每一张合作论文画一条边ーー形成一个电网络，每条边上有一个一欧姆电阻器。两个节点之间的总电阻表示这两个节点的“关闭”程度。

It has been argued that "for an individual researcher, a measure such as Erdős number captures the structural properties of [the] network whereas the h-index captures the citation impact of the publications," and that "One can be easily convinced that ranking in coauthorship networks should take into account both measures to generate a realistic and acceptable ranking."^[29]

It has been argued that "for an individual researcher, a measure such as Erdős number captures the structural properties of [the] network whereas the h-index captures the citation impact of the publications," and that "One can be easily convinced that ranking in coauthorship networks should take into account both measures to generate a realistic and acceptable ranking."

有人认为，”对于单个研究人员来说，erd 数字这样的衡量标准反映了网络的结构特性，而 h-index 则反映了出版物的引用影响” ，”人们很容易相信，在合作网络中的排名应该考虑到这两项措施，以产生一个现实的和可接受的排名

In 2004 William Tozier, a mathematician with an Erdős number of 4, auctioned off a co-authorship on eBay, hence providing the buyer with an Erdős number of 5. The winning bid of $1031 was posted by a Spanish mathematician, who however did not intend to pay but just placed the bid to stop what he considered a mockery.^[30]^[31]

In 2004 William Tozier, a mathematician with an Erdős number of 4, auctioned off a co-authorship on eBay, hence providing the buyer with an Erdős number of 5. The winning bid of $1031 was posted by a Spanish mathematician, who however did not intend to pay but just placed the bid to stop what he considered a mockery.

2004年，数学家威廉 · 托齐尔(William Tozier)在 eBay 上拍卖了一幅共同作者的作品，给买家提供了厄德数为5的作品。一位西班牙数学家以1031美元的价格成交，但是这位数学家并不打算出价，只是为了阻止这个他认为是嘲弄的东西。

Variations

A number of variations on the concept have been proposed to apply to other fields.

对这一概念提出了若干变通办法，以适用于其他领域。

The best known is the Bacon number (as in the game Six Degrees of Kevin Bacon), connecting actors that appeared in a film together to the actor Kevin Bacon. It was created in 1994, 25 years after Goffman's article on the Erdős number.

最著名的是“培根号码”(就像在游戏《凯文 · 培根的六度》中一样) ，它把一起出现在电影中的演员和演员凯文 · 培根联系在一起。它创建于1994年，比戈夫曼关于厄尔德数字的文章晚了25年。

A small number of people are connected to both Erdős and Bacon and thus have an Erdős–Bacon number, which combines the two numbers by taking their sum. One example is the actress-mathematician Danica McKellar, best known for playing Winnie Cooper on the TV series The Wonder Years.

A small number of people are connected to both Erdős and Bacon and thus have an Erdős–Bacon number, which combines the two numbers by taking their sum. One example is the actress-mathematician Danica McKellar, best known for playing Winnie Cooper on the TV series The Wonder Years.

一小部分人同时连接到 erd 和 Bacon，因此有一个 erd s-Bacon 数，它通过求和来组合这两个数。一个例子是女演员兼数学家丹妮卡 · 麦凯勒，她因在电视连续剧《奇迹年代》中扮演温妮 · 库珀而闻名。

Her Erdős number is 4,^[32] and her Bacon number is 2.^[33]

Her Erdős number is 4, and her Bacon number is 2.

她的 erd 数字是4，而她的 Bacon 数字是2。

Further extension is possible. For example, the "Erdős–Bacon–Sabbath number" is the sum of the Erdős–Bacon number and the collaborative distance to the band Black Sabbath in terms of singing in public. Physicist Stephen Hawking had an Erdős–Bacon–Sabbath number of 8,^[34] and actress Natalie Portman has one of 11 (her Erdős number is 5).^[35]

Further extension is possible. For example, the "Erdős–Bacon–Sabbath number" is the sum of the Erdős–Bacon number and the collaborative distance to the band Black Sabbath in terms of singing in public. Physicist Stephen Hawking had an Erdős–Bacon–Sabbath number of 8, and actress Natalie Portman has one of 11 (her Erdős number is 5).

进一步延长是可能的。例如，“ erd s-Bacon-Sabbath 数”是 erd s-Bacon 数和黑色 Sabbath 乐队在公共场合演唱时的合作距离之和。物理学家斯蒂芬 · 霍金的 erd-Bacon-Sabbath 数字是8，女演员娜塔莉 · 波特曼的 erd 数字是11(她的 erd 数字是5)。

References

↑ "Facts about Erdös Numbers and the Collaboration Graph - The Erdös Number Project- Oakland University". wwwp.oakland.edu. Retrieved 2016-10-27.
↑ López de Prado, Marcos (2016). "Mathematics and Economics: A reality check". The Journal of Portfolio Management. 43 (1): 5–8. doi:10.3905/jpm.2016.43.1.005.
↑ "Some Famous People with Finite Erdős Numbers". oakland.edu. Retrieved 4 April 2014.
↑ ^4.0 ^4.1 ^4.2 "List of all people with Erdos number less than or equal to 2". The Erdös Number Project. Oakland University. 14 July 2015. Retrieved 25 August 2015.
↑ Richard Lenski (May 28, 2015). "Erdös with a non-kosher side of Bacon".
↑ Grossman, J. (2015): "The Erdős Number Project." http://wwwp.oakland.edu/enp/erdpaths/
↑ Kishan, Saijel (2016-11-11). "Six Degrees of Quant: Kevin Bacon and the Erdős Number Mystery". Bloomberg.com. Retrieved 2016-11-12.
↑ Bailey, David H. (2016-11-06). "Erdős Numbers: A True "Prince and the Pauper" story". The Mathematical Investor (in English). Retrieved 2016-11-12.
↑ Toby Handfield. "Philosophy research networks".
↑ 引用错误：无效<ref>标签；未给name属性为project的引用提供文字
↑ Erdős, Paul, Robert McEliece, and Herbert Taylor (1971). "Ramsey bounds for graph products" (PDF). Pacific Journal of Mathematics. 37 (1): 45–46. doi:10.2140/pjm.1971.37.45.{{cite journal}}: CS1 maint: multiple names: authors list (link)
↑ White, Douglas R.; Harary, Frank (2001). "The Cohesiveness of Blocks in Social Networks: Node Connectivity and Conditional Density". Sociological Methodology. 31: 305–59. doi:10.1111/0081-1750.00098.
↑ "VITA: Douglas R.White, Anthropology & Social Science Professor, UC-Irvine". Retrieved December 14, 2017.
↑ Barry Wellman, Ove Frank, Vicente Espinoza, Staffan Lundquist and Craig Wilson. "Integrating Individual, Relational and Structural Analysis". 1991. Social Networks 13 (Sept.): 223-50.
↑ Ove Frank; Frank Harary, "Cluster Inference by Using Transitivity Indices in Empirical Graphs." Journal of the American Statistical Association, 77, 380. (Dec., 1982), pp. 835–840.
↑ Erdős, Paul; Marcus, Solomon (1957). "Sur la décomposition de l'espace euclidien en ensembles homogènes" [On the decomposition of the Euclidean space into homogeneous sets]. Acta Mathematica Hungarica. 8: 443–452. doi:10.1007/BF02020326. MR 0095456.
↑ 引用错误：无效<ref>标签；未给name属性为Erdős Number Project的引用提供文字
↑ Grossman, Jerry. "Some Famous People with Finite Erdős Numbers". Retrieved 1 February 2011.
↑ Pachter, L; Batzoglou, S; Spitkovsky, VI; Banks, E; Lander, ES; Kleitman, DJ; Berger, B (1999). "A dictionary-based approach for gene annotation". J Comput Biol. 6: 419–30. doi:10.1089/106652799318364. PMID 10582576.
↑ Kleitman, Daniel. "Publications Since 1980 more or less". Massachusetts Institute of Technology.
↑ von Fintel, Kai (2004). "My Erdös Number is 8". Semantics, Inc. Archived from the original on 23 August 2006.
↑ "Aaron Dinkin has a web site?". Ling.upenn.edu. Retrieved 2010-08-29.
↑ "Mark Liberman's Home Page". Ling.upenn.edu. Retrieved 2010-08-29.
↑ "Christopher Potts: Miscellany". Stanford.edu. Retrieved 2010-08-29.
↑ "Bob's Erdős Number". Lingo.stanford.edu. Retrieved 2010-08-29.
↑ Bowen, Jonathan P.; Wilson, Robin J. (10–12 July 2012). Dunn, Stuart; Bowen, Jonathan P.; Ng, Kia (eds.). Visualising Virtual Communities: From Erdős to the Arts. Electronic Workshops in Computing. British Computer Society. pp. 238–244. {{cite conference}}: Unknown parameter |booktitle= ignored (help)
↑ "Paths to Erdös - The Erdös Number Project- Oakland University". oakland.edu.
↑ Tompa, Martin (1989). "Figures of merit". ACM SIGACT News. 20 (1): 62–71. doi:10.1145/65780.65782. Tompa, Martin (1990). "Figures of merit: the sequel". ACM SIGACT News. 21 (4): 78–81. doi:10.1145/101371.101376.
↑ Kashyap Dixit, S Kameshwaran, Sameep Mehta, Vinayaka Pandit, N Viswanadham, Towards simultaneously exploiting structure and outcomes in interaction networks for node ranking, IBM Research Report R109002, February 2009; also appeared as Kameshwaran, S.; Pandit, V.; Mehta, S.; Viswanadham, N.; Dixit, K. (2010). "Outcome aware ranking in interaction networks" (PDF). Proceedings of the 19th ACM international conference on Information and knowledge management (CIKM '10): 229–238. doi:10.1145/1871437.1871470. ISBN 978-1-4503-0099-5.

↑ Clifford A. Pickover: A Passion for Mathematics: Numbers, Puzzles, Madness, Religion, and the Quest for Reality. Wiley, 2011,

, S. 33 (模板:Google books)

↑ Klarreich, Erica (2004). "Theorem for Sale". Science News. 165 (24): 376–377. JSTOR 4015267.
↑ McKellar's co-author Lincoln Chayes published a paper with Elliott H. Lieb, who in turn co-authored a paper with Daniel Kleitman, a co-author of Paul Erdős.
↑ Danica McKellar was in The Year That Trembled (2002) with James Kisicki, who was in Telling Lies in America (1997) with Kevin Bacon.
↑ Fisher, Len (2016-02-17). "What's your Erdős–Bacon–Sabbath number?". Times Higher Education. Retrieved 2018-07-29.
↑ Sear, Richard (2012-09-15). "Erdős–Bacon–Sabbath numbers". Department of Physics, University of Surrey. Retrieved 2018-07-29.

External links

Jerry Grossman, The Erdős Number Project. Contains statistics and a complete list of all mathematicians with an Erdős number less than or equal to 2.

"On a Portion of the Well-Known Collaboration Graph", Jerrold W. Grossman and Patrick D. F. Ion.

"Some Analyses of Erdős Collaboration Graph", Vladimir Batagelj and Andrej Mrvar.

American Mathematical Society, [1]. A search engine for Erdős numbers and collaboration distance between other authors. As of 18 November 2011 no special access is required.

Numberphile video. Ron Graham on imaginary Erdős numbers.

Number

数目

Category:Social networks

分类: 社交网络

Category:Mathematics literature

类别: 数学文献

Category:Separation numbers

分类: 离职号码

Category:Bibliometrics

分类: 文献计量学

This page was moved from wikipedia:en:Erdős number. Its edit history can be viewed at 埃尔德什数理论/edithistory

[1] "Facts about Erdös Numbers and the Collaboration Graph - The Erdös Number Project- Oakland University". wwwp.oakland.edu. Retrieved 2016-10-27.

[2] López de Prado, Marcos (2016). "Mathematics and Economics: A reality check". The Journal of Portfolio Management. 43 (1): 5–8. doi:10.3905/jpm.2016.43.1.005.

[paths-3] "Some Famous People with Finite Erdős Numbers". oakland.edu. Retrieved 4 April 2014.

[erdos2-4] 4.0 ^4.1 ^4.2 "List of all people with Erdos number less than or equal to 2". The Erdös Number Project. Oakland University. 14 July 2015. Retrieved 25 August 2015.

[5] Richard Lenski (May 28, 2015). "Erdös with a non-kosher side of Bacon".

[6] Grossman, J. (2015): "The Erdős Number Project." http://wwwp.oakland.edu/enp/erdpaths/

[7] Kishan, Saijel (2016-11-11). "Six Degrees of Quant: Kevin Bacon and the Erdős Number Mystery". Bloomberg.com. Retrieved 2016-11-12.

[8] Bailey, David H. (2016-11-06). "Erdős Numbers: A True "Prince and the Pauper" story". The Mathematical Investor (in English). Retrieved 2016-11-12.

[9] Toby Handfield. "Philosophy research networks".

[project-10] 引用错误：无效<ref>标签；未给name属性为project的引用提供文字

[11] Erdős, Paul, Robert McEliece, and Herbert Taylor (1971). "Ramsey bounds for graph products" (PDF). Pacific Journal of Mathematics. 37 (1): 45–46. doi:10.2140/pjm.1971.37.45.{{cite journal}}: CS1 maint: multiple names: authors list (link)

[12] White, Douglas R.; Harary, Frank (2001). "The Cohesiveness of Blocks in Social Networks: Node Connectivity and Conditional Density". Sociological Methodology. 31: 305–59. doi:10.1111/0081-1750.00098.

[13] "VITA: Douglas R.White, Anthropology & Social Science Professor, UC-Irvine". Retrieved December 14, 2017.

[14] Barry Wellman, Ove Frank, Vicente Espinoza, Staffan Lundquist and Craig Wilson. "Integrating Individual, Relational and Structural Analysis". 1991. Social Networks 13 (Sept.): 223-50.

[15] Ove Frank; Frank Harary, "Cluster Inference by Using Transitivity Indices in Empirical Graphs." Journal of the American Statistical Association, 77, 380. (Dec., 1982), pp. 835–840.

[16] Erdős, Paul; Marcus, Solomon (1957). "Sur la décomposition de l'espace euclidien en ensembles homogènes" [On the decomposition of the Euclidean space into homogeneous sets]. Acta Mathematica Hungarica. 8: 443–452. doi:10.1007/BF02020326. MR 0095456.

[Erd.C5.91s_Number_Project-17] 引用错误：无效<ref>标签；未给name属性为Erdős Number Project的引用提供文字

[18] Grossman, Jerry. "Some Famous People with Finite Erdős Numbers". Retrieved 1 February 2011.

[19] Pachter, L; Batzoglou, S; Spitkovsky, VI; Banks, E; Lander, ES; Kleitman, DJ; Berger, B (1999). "A dictionary-based approach for gene annotation". J Comput Biol. 6: 419–30. doi:10.1089/106652799318364. PMID 10582576.

[20] Kleitman, Daniel. "Publications Since 1980 more or less". Massachusetts Institute of Technology.

[21] von Fintel, Kai (2004). "My Erdös Number is 8". Semantics, Inc. Archived from the original on 23 August 2006.

[22] "Aaron Dinkin has a web site?". Ling.upenn.edu. Retrieved 2010-08-29.

[23] "Mark Liberman's Home Page". Ling.upenn.edu. Retrieved 2010-08-29.

[24] "Christopher Potts: Miscellany". Stanford.edu. Retrieved 2010-08-29.

[25] "Bob's Erdős Number". Lingo.stanford.edu. Retrieved 2010-08-29.

[26] Bowen, Jonathan P.; Wilson, Robin J. (10–12 July 2012). Dunn, Stuart; Bowen, Jonathan P.; Ng, Kia (eds.). Visualising Virtual Communities: From Erdős to the Arts. Electronic Workshops in Computing. British Computer Society. pp. 238–244. {{cite conference}}: Unknown parameter |booktitle= ignored (help)

[27] "Paths to Erdös - The Erdös Number Project- Oakland University". oakland.edu.

[28] Tompa, Martin (1989). "Figures of merit". ACM SIGACT News. 20 (1): 62–71. doi:10.1145/65780.65782. Tompa, Martin (1990). "Figures of merit: the sequel". ACM SIGACT News. 21 (4): 78–81. doi:10.1145/101371.101376.

[Dixit-29] Kashyap Dixit, S Kameshwaran, Sameep Mehta, Vinayaka Pandit, N Viswanadham, Towards simultaneously exploiting structure and outcomes in interaction networks for node ranking, IBM Research Report R109002, February 2009; also appeared as Kameshwaran, S.; Pandit, V.; Mehta, S.; Viswanadham, N.; Dixit, K. (2010). "Outcome aware ranking in interaction networks" (PDF). Proceedings of the 19th ACM international conference on Information and knowledge management (CIKM '10): 229–238. doi:10.1145/1871437.1871470. ISBN 978-1-4503-0099-5.

[30] Clifford A. Pickover: A Passion for Mathematics: Numbers, Puzzles, Madness, Religion, and the Quest for Reality. Wiley, 2011,
v
t
e
Inline tags
Attribution

{{Attribution needed}}

{{By whom}}

{{Weasel-inline}}

{{Which?}}

{{Who}}

Citation

{{Better source}}

{{Citation needed}}
{{Citation needed span}}

{{Reference necessary}}

{{Citation needed (lead)}}

{{Failed verification}}

{{Full}}

{{Primary source-inline}}

{{Request quotation}}

{{Retracted}}

{{Third-party-inline}}

Incomplete

{{Author missing}}

{{Author incomplete}}

{{Date missing}}

{{ISBN missing}}

{{Page needed}}

{{Publisher missing}}

{{Title incomplete}}

{{Year missing}}

{{Dead link}}

Content

{{Contradict-inline}}

{{Contradiction-inline}}

{{Dubious|talk page section name}}

{{Examples}}

{{Inconsistent}}

{{List fact}}

{{Lopsided}}

{{POV-statement}}

Date and place

{{Clarify timeframe}}

{{Quantify}}

{{Update after}}

{{Update-small}}

{{When}}

{{Where}}

{{Year needed}}

Wikification

{{Disambiguation needed}}

{{Pronunciation needed}}

Wording

{{Ambiguous}}

{{Awkward}}

{{Buzz}}

{{Clarify}}

{{Clarify span}}

{{Definition}}

{{Elucidate}}

{{Expand acronym}}

{{Technical-statement}}

{{Vague|optional message to be displayed on mouseover}}

{{Why?}}

Category
, S. 33 (模板:Google books)

[31] v

[32] t

[33] e

[34] {{Attribution needed}}

[35] {{By whom}}

[36] {{Weasel-inline}}

[37] {{Which?}}

[38] {{Who}}

[39] {{Better source}}

[40] {{Citation needed}}
{{Citation needed span}}

{{Reference necessary}}

[41] {{Citation needed span}}

[42] {{Reference necessary}}

[43] {{Citation needed (lead)}}

[44] {{Failed verification}}

[45] {{Full}}

[46] {{Primary source-inline}}

[47] {{Request quotation}}

[48] {{Retracted}}

[49] {{Third-party-inline}}

[50] {{Author missing}}

[51] {{Author incomplete}}

[52] {{Date missing}}

[53] {{ISBN missing}}

[54] {{Page needed}}

[55] {{Publisher missing}}

[56] {{Title incomplete}}

[57] {{Year missing}}

[58] {{Dead link}}

[59] {{Contradict-inline}}

[60] {{Contradiction-inline}}

[61] {{Dubious|talk page section name}}

[62] {{Examples}}

[63] {{Inconsistent}}

[64] {{List fact}}

[65] {{Lopsided}}

[66] {{POV-statement}}

[67] {{Clarify timeframe}}

[68] {{Quantify}}

[69] {{Update after}}

[70] {{Update-small}}

[71] {{When}}

[72] {{Where}}

[73] {{Year needed}}

[74] {{Disambiguation needed}}

[75] {{Pronunciation needed}}

[76] {{Ambiguous}}

[77] {{Awkward}}

[78] {{Buzz}}

[79] {{Clarify}}

[80] {{Clarify span}}

[81] {{Definition}}

[82] {{Elucidate}}

[83] {{Expand acronym}}

[84] {{Technical-statement}}

[85] {{Vague|optional message to be displayed on mouseover}}

[86] {{Why?}}

[31] Klarreich, Erica (2004). "Theorem for Sale". Science News. 165 (24): 376–377. JSTOR 4015267.

[32] McKellar's co-author Lincoln Chayes published a paper with Elliott H. Lieb, who in turn co-authored a paper with Daniel Kleitman, a co-author of Paul Erdős.

[33] Danica McKellar was in The Year That Trembled (2002) with James Kisicki, who was in Telling Lies in America (1997) with Kevin Bacon.

[34] Fisher, Len (2016-02-17). "What's your Erdős–Bacon–Sabbath number?". Times Higher Education. Retrieved 2018-07-29.

[35] Sear, Richard (2012-09-15). "Erdős–Bacon–Sabbath numbers". Department of Physics, University of Surrey. Retrieved 2018-07-29.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

[27]

[28]

[29]

[30]

[31]

[32]

[33]

[34]

[35]

@@ 第116行： / 第116行： @@
 ! Co-author !! Number of <br>collaborations
-!合著者! ！合作次数
 |-
@@ 第152行： / 第150行： @@
 | Ralph Faudree || 50
-拉尔夫 · 法德里 | 50
 |-
@@ 第176行： / 第172行： @@
 | Cecil C. Rousseau || 35
-塞西尔 · c · 卢梭 | 35
 |-
@@ 第236行： / 第230行： @@
 | Ronald Graham || 28
-罗纳德 · 格雷厄姆28
 |}