Feature Column

Pseudonyms in Mathematics

Why would there be pseudonyms for the publication of research papers in mathematics? One possible motive might be... that a group of people collaborated on a piece of work and wanted to simplify the multiple authorship to a single name. Another reason appears to be playfulness or whimsy.

Joseph Malkevitch
York College (CUNY)
Email Joseph Malkevitch

 

Introduction

 

Are you familiar with the novels of Currer Bell or Ellis Bell? Many people who have enjoyed reading Jane Eyre (1847) and Wuthering Heights (1848) are unaware that the authors of these books at the time they were published were known as Currer Bell and Ellis Bell.

In the 19th century there were several examples of ultimately famous women writing under a male pseudonym. Among the most notable were George Eliot as the name for Mary Ann Evans and George Sands as the name for Amantine Lucile Aurore Dupin. (Eliot attended courses on mathematics and physics in London and Geneva and incorporated mathematical themes in her novels.)

Portrait of George Eliot

Figure 1 (George Eliot, aka Mary Ann Evans, as painted by Alexandre-Louis-François d'Albert-Durade. Image courtesy of Wikipedia.)
 

Few who have not studied English literature will recognize the names of Currer Bell, Ellis Bell and Acton Bell as the names under which Charlotte (1816-1855), Emily (1818-1848) and Anne (1820-1849) Brontë published their work in their own times! Another interesting example, this time of a man writing under a pseudonym in the 19th century, was Lewis Carroll, whose true name was C.L. Dodgson.

Lewis Carroll

Figure 2 (C.L. Dodgson, otherwise known as Lewis Carroll, courtesy of Wikipedia.)
 

Dodgson was a distinguished mathematician but not as distinguished as he was to become by being an author! Few people know Dodgson for his mathematics but many have read the book or seen the movie version of Alice in Wonderland! In recent times, Dodgson's mathematical fame has been revived in association with his appealing method of using ranked ballots to decide elections.

While not a pseudonym, the family name that James Joseph Sylvester (1814-1897) was born into was "Joseph," not Sylvester. The Sylvester family practiced Judaism and Sylvester's older brother decided to change the family name from Joseph to Sylvester to avoid some of the anti-semitism that plagued British Jews at the time.

James Sylvester

Figure 3 (James Joseph Sylvester, courtesy of Wikipedia)
 

Sylvester was a student at Cambridge but when it came time to graduate and to get his Master's degree regulations would have required him to sign the "Articles," pledging allegiance to the Church of England, something he refused to do. It was his good fortune that Trinity College in Dublin was willing to grant him his degree based on his work at Cambridge because they were more open minded about religion! Ireland was a country dominated by Catholicism at the time while Britain had an established church because Henry VIII had broken with Catholicism and set in motion years of religious turmoil by bringing a Protestant reform movement to England. However, a big part of his motive was that the Vatican would not allow him to divorce his first wife, Catherine of Aragon, who was Catholic so that he could marry Anne Boleyn, who gave birth to the woman who would become Queen Elizabeth I!

Women who wanted to pursue mathematics, or more generally an education, had limited opportunities in the 19th century. It was not until 1869 that Girton College at Cambridge admitted women and, thus, offered a path for women who wanted to pursue mathematics to study the subject in a university setting. Winifred Edgerton Merrill became the first American woman to earn a PhD in mathematics, which she earned in 1886 from Columbia University. But rather amazingly, the granting of degrees to women by Cambridge was not allowed until 1948. The situation for women and Jews had similarities. Despite his growing reputation as a mathematician, Sylvester was not able to get an appointment at Oxford because of his religion. His famous stays in America were attempts to have his talents recognized by academic institutions but for complex reasons, some of which seem to have been related to his personality, he was unable to get a stable long term position in the United States. Ironically, he returned from America to Britain because he was offered the Savilian Professorship in Geometry at Oxford University, because the rules forbidding Jews to have that position had changed. (For more about Sylvester's career and the careers of prominent women mathematicians such as Emmy Noether and Sofya Kovalevskaya, the first woman to earn a doctorate in mathematics, see the April 2015 Feature Column on Mathematical Careers.)

Sofya Kovalevskaya

Figure 4 (Sofya Kovalevskaya, courtesy of Wikipedia.)
 

Why would there be pseudonyms for the publication of research papers in mathematics? One possible motive might be that a research paper might seem to a community used to a particular person's "impressive" work, to be less impressive and one might want to protect one's reputation by publishing under a pseudonym. Another reason might be that a group of people collaborated on a piece of work and wanted to simplify the multiple authorship to a single name. Another reason appears to be playfulness or whimsy. While there are many pseudonyms that were used for groups of mathematicians, the best known is Bourbaki. Bourbaki represented a group of mathematicians, initially primarily operating out of France, but having members eventually from many countries, to put mathematics on a "sound" foundation rather than look at its separate pieces as the total edifice. Bourbaki stood for the idea of showing with "hindsight" how to structure the vast river of mathematical knowledge in a way that built organically from a well structured foundation. I will not attempt to look at the members of Bourbaki here or its many accomplishments. Many of the members of Bourbaki had distinguished careers as individuals. I will not take on the history and mathematical accomplishments of Bourbaki, not because they are vast, though they are, but in favor of pseudonyms for groups who work in mathematical areas that have good track records with looking at mathematics that can be understood with a relatively limited mathematics background. My goal here is to mention two relatively recent groups of mathematicians who published under an anonymous name. I will say some things about why they might have done this and call attention to some of the research work that as individuals these people accomplished.

So I will look at in turn look at the work of Blanche Descartes and G.W. Peck. Blanche started "her" work earlier that "G.W.," so I will start with her.

 

Blanche Descartes

 

What associations do you have with the name Blanche Descartes? Perhaps Blanche Descartes was Rene Descartes's (1596-1650) wife or daughter? In fact Descartes never married but he did have a daughter Francine who died at the age of 5. But seemingly the name Blanche Descartes has no direct connection with Descartes, the great French mathematician and philosopher. Blanche Descartes was a pseudonym initially for joint work by Rowland Brooks, Cedric Smith, Arthur Stone and William Tutte. The four of them met as undergraduate students at Cambridge University in 1936 and were members of the Trinity Mathematical Society. The photo below shows them attending a meeting of the Trinity Mathematical Society, which as you notice, has no female members. Female students were not allowed to study at Trinity College until much later, the first women arriving in 1976!

Trinity Mathematical Society

Figure 5 (The Trinity Mathematical Society, photo courtesy of Wikipedia.)
 

Trinity College at Cambridge, where Isaac Newton had been an undergraduate and later a Fellow, was especially associated with mathematics. The four decided to use the name Blanche Descartes to publish some initial joint work they did. It has been argued that "Blanche" derives from letters involved in their names. Using first letters of Bill (William), Leonard (Brooks middle name), Arthur and Cedric, one gets BLAC and with some "padding" this becomes: BLAnChe. Perhaps Descartes involves a play on the phrase 'carte blanche' and since they all were studying mathematics for different reasons, perhaps a reference to Rene Descartes, who also dabbled in things other than mathematics.

One early question that the four looked at has to do with a problem which has come to be known as "squaring the square." The problem involves taking a square and decomposing (tiling) it into smaller squares of different side lengths. One can also consider the more general question of when a rectangle can be tiled with squares of different lengths. Figure 6 shows an example, discovered by A. Duijvestijn.

A squared square

Figure 6 (A square decomposed into squares of all different sizes. This one has the smallest number of possible squares in any such decomposition. Image courtesy of Wikipedia.)
 

This tantalizing question attracted much interest over the years and has been shown to have unexpected connections to mathematical ideas that seem to have no connection with tilings. Some interest in this problem occurred in the "recreational" mathematics community. It is not exactly clear why Brooks, Smith, Stone and Tutte chose to use a pseudonym but perhaps it might have had to do with their concern that this question was of less mathematical "importance" than topics their teachers at Cambridge considered worth students investing time in investigating. Mathematics is full of examples that were considered recreational at one time but have been shown to be rooted in deeper mathematical ideas. The squared-square problem is an example of such a problem.

For each of the members of Blanche Descartes I will indicate some information about them and try to provide an example of some mathematics associated with them. All of them were to achieve some measure of fame and acclaim under their "real" names, sometimes in limited venues but in Tutte's case on a big stage.

 

William Tutte (1917-2002)

 

William Tutte

Figure 7 (Photo of William Tutte. Courtesy of Wikipedia.)
 

Although Tutte was born in England, his academic career was spent largely in Canada at the University of Waterloo. This university has a complex footprint of the presence of mathematics:

  • Department of Applied Mathematics.
  • Department of Combinatorics and Optimization.
  • David R. Cheriton School of Computer Science.
  • Department of Pure Mathematics.
  • Department of Statistics and Actuarial Science.

When Tutte first went to Canada, he did so at the invitation of H.S.M. Coxeter to join the faculty at the University of Toronto. Not long after the University of Waterloo was founded, Tutte moved there and was associated with the Department of Combinatorics and Optimization which at the present time has about 30 members. The reason why his work in England before taking up residency at Waterloo is less well known is that he was part of the team that was assembled in England at Bletchley Park. Even after the war the people who worked at Bletchley Park were not allowed to speak of their work there until quite recently. This team included Alan Turing, who is justly famous for his work on breaking the Enigma machine cipher. However, even after the Enigma ciphers were being read, a code known as Fish, used by the German army and for messages sent by leaders in Berlin to field commanders, had not been broken. Tutte was one of a team that managed to read Fish even though, unlike the case for Enigma, there was no physical machine in the possession of the code breakers to help them with breaking the code. In essence Tutte managed to figure out how the machine used to send messages in Fish worked by reconstructing the exact operation of the machine even without having seen the machine! When this was accomplished, it became possible to decipher Fish messages for the remainder of the war without the Germans realizing that their codes were being read. This work was done before Tutte completed his doctorate.

When WWII ended, Tutte returned as a graduate student to Cambridge in 1945 to work on his doctorate. During this period he produced an example related to the famous four-color theorem which is known today as the Tutte graph in his honor. In this context, a graph is a collection of points (called vertices) and edges, which can be curved or straight, that join up vertices. There are various definitions of graphs but the one preferred by Tutte allows vertices to be joined to themselves (these edges are called loops) and allows pairs of distinct vertices to be joined by more than one edge. We see in Figure 8 on the right the Tutte graph, put together using three copies of the configuration on the left, known as a Tutte triangle. This graph has three edges at every vertex, that is, it is 3-valent or degree 3, is planar (can be drawn in the plane so that edges meet only at vertices) and is three-connected. When a graph has been drawn in the plane so that edges meet only at vertices, the graph is called a plane graph. 3-connected means that for any two vertices $u$ and $v$ in the graph one can find at least 3 paths from $u$ to $v$ that have only the vertices $u$ and $v$ in common. What is special about the Tutte graph is that there is no simple tour of its vertices that visits each vertex once and only once in a simple circuit, a tour usually known as a Hamilton circuit. If it were true that every 3-valent 3-connected graph in the plane had a hamilton circuit there would be a simple proof of the four-color theorem (every graph drawn in the plane can be face colored with 4 or fewer colors). (The coloring rule is that any two faces that share an edge would have to get different colors.)

A Tutte triangle The non-hamiltonian Tutte graph

Figure 8 (A Tutte triangle on the left and three Tutte triangles assembled to form a plane, 3-valent, 3-connected graph with no Hamilton circuit. Images courtesy of Wikipedia.)
 

While the Tutte graph is not the smallest planar, degree 3, 3-connected graph which has no simple closed curve tour of all of its vertices (such a tour is called a Hamilton circuit or HC), it is relatively easy to see why it can't have an HC. This fact follows from that one can show that if an HC visits the vertices of a Tutte triangle then the HC must enter using one of the two edges shown at the bottom and exit the Tutte triangle using the edge at the top. For those interested in applications of mathematics, the problem known as the Traveling Salesperson Problem (TSP) requires finding a minimum cost Hamilton circuit in a graph which has non-zero weights on its edges. Finding routes for school buses, group taxis, or when you run errands involves problems of this kind. Finding solutions to the TSP for large graphs seems to require rapidly growing amounts of computation as the number of sites to be visited (the vertices of the graph) grows.

In addition to his famous example of a graph not having a Hamilton circuit Tutte also showed a very appealing theorem about when a graph must have a Hamilton circuit.:

Theorem (W. T. Tutte).
Every 4-connected plane graph has a hamiltonian circuit.

The 4-connected condition means that given any pair of vertices $u$ and $v$ in the graph there must be at least 4 paths from $u$ to $v$ that have only $u$ and $v$ in common. In particular, this means that the graph must have at least 4 edges at each vertex. Figures show a very regular 4-connected plane graph the graph of the Platonic Solid known as the icosahedron and a much more irregular graph. Can you find a Hamilton circuit in each of these graphs? While there are algorithms (procedures) for finding an HC in a plane 4-connected graph that run relatively fast, the problem of finding an HC in an arbitrary graph is known to be computationally hard.

Graph of the icosahedron

Figure 9 (A 4-connected planar graph. This is the graph of the regular icosahedron. Image courtesy of Wikipedia.)
 
A plane 4-connected graph

Figure 10 (A planar 4-connected graph which is not very symmetric.)
 

 

Cedric A. B. Smith (1917-2002)

 

While Smith studied mathematics at Cambridge and published some research work in mathematics, his career path did not involve becoming an academic mathematician associated with a mathematics department. Rather he pursued a career as a statistician and worked at the Galton Laboratory of University College, University of London. Some of Smith's work was related to genomics, helping to locate where particular genes were on a chromosome.

Like Tutte, Smith made an easy-to-state contribution to the theory of Hamilton circuits. Clearly, having conditions that show a family of graphs must have an HC or providing examples of graphs where no HC exists is also intriguing. But it is also natural to ask if there might be families of graphs where graphs might have exactly one HC. Smith was able to provide an answer to this question for the important class of graphs where every vertex has exactly three edges at a vertex.

Theorem.
Every 3-valent cubic graph has a even number of circuits that pass through each vertex once and only once (Hamiltonian circuit).

Because this graph has one Hamilton circuit tour...

A cubic graph with an HC highlighted

Figure 11 (The dark edges show a Hamilton circuit is a 3-valent (cubic) graph.)
 

...by Smith's Theorem it must have another!

 

Rowland Leonard Brooks (1916-1993)

 

Rowland Brooks is most well known for a theorem about coloring graphs that is named for him. Let me begin with an application and show how the theorem of Brooks makes it possible to get insight into such situations.

Suppose we have a collection of committees, say, committees of students and faculty at a college. The collection of individuals who are on some committee will be denoted by S. The committee names are a, b, ..., h and a table (Figure 12) indicates with an X when a pair of committees has one or more members of S on both committees. It would be nice to be able to schedule the 8 committees into as few hourly time slots as possible so that no two committees that have a common member meet at the same time. Committees with no common members could meet at the same time. A person can't be at the meeting of two committees the person serves on if those committees meet at the same time.

A table of committee conflicts data

Figure 12 (A table where when two committees have one or more people on both committees these committees should not meet at the same time. A conflict is indicated with an X.)
 

We can display the conflict information geometrically using a graph by having a dot for each committee and joining two vertices representing committees with an edge if these committees can't meet at the same time because they have members in common.

A graph coding the previous committee conflict data

Figure 13 (A conflict graph for 8 committees. Committees joined by an edge should have their meetings at different times. The graph can be vertex colored with 4 colors.)
 

The vertex coloring problem for graphs assigns a label to each vertex, called a color, with the requirement that two vertices jointed by an edge get different colors. We could assign a different color to each dot (vertex) in Figure 13, which would mean that we would use 8 time slots for the committee meetings. The minimum number of colors needed to color the vertices of a graph is called the chromatic number of a graph. Brooks was able to find a way to get what is often a good estimate for the chromatic number of a graph.

Brook's Theorem:
If G is a graph which is not a complete graph or a circuit of odd length, then the chromatic number of G is at most (less than or equal to) the degree (valence) equal to the largest degree (valence) that occurs in G.

For a general graph it is a difficult computational problem to determine the chromatic number, but Brook's Theorem often allows one to get a good estimate quickly. In particular, if a graph has a million vertices, no matter how complex the graph, if it has maximum valence 5, it will not require more than 5 colors (though the chromatic number may be less).

Can you find a coloring of the vertices in the graph in Figure 13 that improves over the estimate given by Brook's Theorem?

 

Arthur Stone (1916-2000)

 

Arthur Stone eventually made his way to the United States and taught for many years at the University of Rochester. His wife Dorothy (Dorothy Maharam) was also a mathematician and his son David (deceased 2014) and daughter Ellen also became mathematicians.

While Stone contributed to mathematics in a variety of theoretical ways perhaps he is best known for his invention of what are known as flexagons. The hexaflexagon shown in Figure 14 can be folded from the triangles shown in Figure 15.

A hexaflexagon

Figure 14 (An example of a hexaflexagon, image courtesy of Wikipedia.)
 
A template of triangles which fild to a hexaflexagon

Figure 15 (A template for a hexaflexagon, image courtesy of Wikipedia.)
 

While the members of Blanche Descartes published several articles, most of these publications were in fact the work of William Tutte. A more recent example of a group of mathematicians who published together under one name will now be treated.

 

G.W. Peck

 

G. W. Peck's publications can be found here. Unlike Blanche Descartes, whose members are now all deceased, of the 6 mathematicians who have sometimes written under the pseudonym of G.W. Peck, three of the six people involved are still alive (2020). What associations do you have with this name? Perhaps you might think of the distinguished actor Gregory Peck, but Eldred Gregory Peck (April 5, 1916 – June 12, 2003), the actor, did not write under the name G.W. Peck. The individuals involved in G.W. Peck's publications were Ronald Graham, Douglas West, George Purdy, Paul Erdős, Fan Chung, and Daniel Kleitman.

Let me make a few brief comments about each of the members of this remarkable collaboration of distinguished mathematicians in turn. As individuals they illustrate the remarkably varied ways that individuals have mathematical careers.

 

Ron (Ronald) Graham (1935-2020)

 

Ronald Graham

Figure 16 (A photo of Ronald Graham, courtesy of Wikipedia.)
 

Some people perhaps know Ron Graham best for his juggling and studying mathematical problems associated with juggling. However, before his recent death he was one of America's best known research mathematicians. Some of Graham's fame beyond the mathematics community itself was that Martin Gardner, a prolific popularizer of mathematics, often wrote about ideas which were called to his attention by Ron Graham. But somewhat unusually for mathematicians known for their theoretical work, Ron Graham also had a career as a distinguished applied mathematician. He had a long career as a mathematician at Bell Laboratories where he worked at times under the direction of Henry Pollak, who in addition to his work at Bell Laboratories also was a President of NCTM and MAA. Graham during his career was President of the American Mathematical Society and MAA. He also was head of the Mathematics Section (the position no longer exists) of the New York Academy of Sciences. During his long stay at Bell Laboratories he championed examples of discrete mathematical modeling. This included popularizing the use of graphs and digraphs to solve problems involved in scheduling machines in an efficient manner. He also called attention to the way that problems about packing bins with weights had connections to machine scheduling problems.

Graham published on a vast array of topics ranging from very technical to expository articles. His many contributions to mathematics can be sampled here.

 

Douglas West (1953- )

 

Douglas West

Figure 17 (A photo of Douglas West, courtesy of Wikipedia.)
 

In addition to his many research papers, West has published several comprehensive books in the areas of discrete mathematics and combinatorics. He is also noteworthy in maintaining and collecting lists of unsolved problems in the general area of discrete mathematics. In recent years he has helped edited the Problems Section of the American Mathematical Monthly.

 

George Purdy (1944-2017)

 

Purdy made many contributions to discrete geometry. He was particularly interested in questions related to arrangements of lines and the Sylvester-Gallai Theorem.

 

Paul Erdős (1913-1996)

 

The best known author of this group was Paul Erdős (1913-1996) whose career was marked by not having a specific educational or industrial job. For much of his life Erdős traveled between one location and another to work privately with individuals, and/or give public lectures where he often promoted easy-to-state but typically hard-to-solve problems in geometry, combinatorics and number theory.

Paul Erdos

Figure 18 (A photo of Paul Erdős, courtesy of Wikipedia.)
 

Here is a sample problem in discrete geometry that Paul Erdős called attention to, whose investigation over the years has sprouted many new techniques and ideas. It has also resulted in new questions as some aspect of the initial problem got treated. Suppose that $P$ is a finite set of $n$ distinct points in the plane with the distance between the points being computed using Euclidean distance. One must be careful to specify what distance is to be used because, for example, one could instead compute the result using another distance function, say, taxicab distance. Suppose $D(n)$ denotes the set of numbers one gets for distances that occurs between pairs of points in $P$. Paul Erdős raises questions such as:

  1. What is the largest possible number of elements in $D(n)$?
  2. What is the smallest possible number of elements in $D(n)$?
  3. Can all of the numbers in $D(n)$ be integers?
  4. Can all of the integers in $D(n)$ be perfect squares?

Such problems have come to be known as distinct distance questions. Question (a) is quite easy. However, the question (b) is not fully understood even today. Initially Erdős was concerned with the behavior of the smallest number of values that could occur in D(n) as n got larger and larger. As tools for getting insights into problems has grown as well as interesting examples with particular properties, many new questions have come to be asked. For example. if the points of the original set lie in equal numbers on two disjoint circles what is the smallest number of distinct distances that can occur?

Here is a problem in this spirit to contemplate. In each of Figures 19 and 20 we have a configuration of 20 points, together with some of the line segments that join up these points, when we think of the diagrams as graphs. Figure 19 shows a polyiamond, a configuration of equilateral triangles that meet edge to edge while Figure 20 shows a polyomino, congruent squares which meet edge to edge.

A polyiamond with 20 vertices

Figure 19 (A polyiamond with 20 vertices.)
 
A polyomino with 20 vertices

Figure 20 (A polyomino with 20 vertices)
 

Question: Investigate the number of distinct distances and integer distances that can occur in each of the configurations in Figures 19 and 20. Can you formulate interesting questions to ask about distances based on these two configurations and your thinking about them?

 

Fan Chung (1949- )

 

Paul Erdös, Ronald Graham and Fan Chung

Figure 21 (A photo of Ron Graham, Paul Erdős and Fan Chung, photo courtesy of Wikipedia.)
 

Fan Chung is an extremely distinguished mathematician. She was born in Taiwan but did her graduate studies in America, and Herbert Wilf was her doctoral thesis advisor. She was the wife of Ronald Graham, with whom she wrote many joint papers. Like her husband, she worked in both the industrial world and in academia. She worked at Bell Laboratories and at Bellcore after the breakup of the Bell System. Her research covers an unusually broad range of topics, but includes especially important work in number theory, discrete mathematics, and combinatorics. Fan Chung and Ron Graham both made important contributions to Ramsey Theory. Fan Chung still teaches at the University of California, San Diego.

 

Daniel Kleitman (1934- )

 


Daniel Kleitman

Figure 22 (A photo of Daniel Kleitman, courtesy of Wikipedia.)
 

The person who perhaps has most often published under the G.W. Peck moniker has been Daniel Kleitman, whose long career has been associated with MIT. Many of his publications are in the area of combinatorics and discrete mathematics, in particular results about partially ordered sets.

 

Concluding Thoughts

 

To the extent that there are individuals who can't study or publish mathematics because of discrimination, hopefully we can all work to tear down these barriers. Mathematical progress requires all the help it can get and to deny people the fulfillment that many who get pleasure from doing mathematics have is most unfortunate. While in the past pseudonyms were sometimes chosen to avoid prejudicial treatment, we can hope this will seem less and less necessary in the future.

 

References

 

Those who can access JSTOR can find some of the papers mentioned above there. For those with access, the American Mathematical Society's MathSciNet can be used to get additional bibliographic information and reviews of some of these materials. Some of the items above can be found via the ACM Digital Library, which also provides bibliographic services.

  • Appel, K., and W. Haken, J. Koch, Every Planar Map is Four Colorable. I: Discharging. Illinois J. Math. 21 (1977) 429-490.
  • Appel, K. and Haken, W. 1977 Every Planar Map is Four-Colorable, II: Reducibility. Illinois J. Math. 21, 491-567.
  • Ball, Derek Gordon. Mathematics in George Eliot's Novels. PhD thesis, University of Leicester, 2016.
  • Brooks, R., and C. Smith, A. Stone, W. Tutte, The Dissection of Rectangles into Squares, Duke Mathematical Journal, 7 (1940) 312-40.
  • Chiba, N. and T. Nishizeki, "The hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs." Journal of Algorithms 10.2 (1989): 187-211.
  • Chung, Fan RK, Lectures on spectral graph theory, CBMS Lectures, Fresno, 92 (1996): 17-21.
  • Chung, Fan, and R. Graham, Sparse quasi-random graphs, Combinatorica 22 (2002): 217-244.
  • Descartes, Blanche, Network colorings, Math. Gaz. 32 (1948) 67-69.
  • Erdős, .P, and A. H. Stone. On the structure of linear graphs, Bull. Amer. Math. Soc 52.(1946) 1087-1091.
  • Erdős, Paul, and George Purdy. "Some extremal problems in geometry." Discrete Math. 1974.
  • Graham, R., The Combinatorial Mathematics of Scheduling, Scientific American 238 (1978), 124-132.
  • Graham, R., Combinatorial Scheduling Theory, in Mathematics Today: Twelve Informal Essays, ed. L. Steen, Springer-Verlag, N.Y. (1978), 183-211.
  • Henle, M. and B. Hopkins, eds. Martin Gardner in the twenty-first century. Vol. 75. American Mathematical Soc., Providence, 2012.
  • Kleitman, D.. On an extremal property of antichains in partial orders. The LYM property and some of its implications and applications, Combinatorics, 2 (1974) 77-90.
  • Kleitman, Daniel J., and Douglas B. West. "Spanning trees with many leaves." SIAM Journal on Discrete Mathematics 4.1 (1991): 99-106.
  • Purdy, George. "Two results about points, lines and planes." Discrete mathematics 60 (1986): 215-218.
  • Robertson, N. and D. Sanders, P. Seymour, R. Thomas, New Proof of the Four Colour Theorem. Electronic Research Announcement, Amer. Math. Soc. 2, (1996) 17-25.
  • Solymosi, J, and C.. Tóth, Distinct distances in the plane, Discrete & Computational Geometry 25 (2001): 629-634.
  • Smith, C. and S Abbott, The story of Blanche Descartes. Mathematical Gazette (2003) 23-33.
  • Stone, A. H., Paracompactness and product spaces, Bulletin of the American Mathematical Society 54 (1948): 977-982.
  • Szekely, Crossing numbers and hard Erdős problems in discrete geometry, Combinatorics, Probability and Computing 6 (1997): 353-358.
  • Tutte, W. T., Squaring the Square, Mathematical Games column, ed. M. Gardner, Scientific American, Nov. 1958.
  • West, D., Introduction to Graph Theory. Vol. 2. Upper Saddle River, NJ: Prentice hall, 1996.

Joseph Malkevitch
York College (CUNY)
Email Joseph Malkevitch


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