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# No Strangers at This Party

## Introduction

In this article I attempt to address the following general question:

What should an instructor do when they have an opportunity to expose a group of undergraduate students, who are not necessarily planning to become professional mathematicians, to a set of advanced mathematical ideas?

Since 2014, I have been assigned to teach Simon Fraser University’s Mathematical Journeys course four times. The catalogue description of this undergraduate course, with a topic that varies from term to term, explains that “each [journey] is designed to appeal particularly to mathematics minor students and others with a broad interest in mathematics.”

In other words, this course is for undergraduates who enjoy mathematics but may have chosen another academic program as their main field of study. The purpose of offering such a course is to expose students, at an appropriate level, to mathematical ideas and the applications of mathematics that are commonly not part of the undergraduate curriculum.

My topic choice has been Ramsey theory, a dynamic segment of contemporary mathematics that combines, among other concepts, ideas from number theory and combinatorics.

There are three main reasons for this choice:

- •
Mathematical: Some of the fundamental Ramsey theory results, like Ramsey’s theorem or van der Waerden’s theoremFootnote

^{1}, are self-contained. As such, they are accessible to mathematically curious students willing to explore big ideas.- •
Educational: The story about the beginnings and the development of Ramsey theory and people involved, like Frank P. Ramsey or Paul Erdős, for example, supports the view of mathematics as a living organism, i.e., something that grows and adapts, and a deeply human endeavor.

- •
Personal: I was privileged to witness how Ramsey theory served as a portal into the world of mathematical research for some extraordinarily talented young mathematicians. My hope was that I might illuminate this portal to at least some of my students.

In short, the main goal of the course was to provide students with a very gentle introduction to a set of advanced mathematical ideas and problems, their history, and their place in modern mathematics. In addition, my wish was to give my students both challenging and enjoyable learning experiences.

The major challenge that I, as a course instructor, faced was the diversity of my students’ academic backgrounds. For example, in the fall 2020, my 62 students were enrolled in 17 different academic programs ranging from first nation studies to education to computing science to mathematics. It felt as though that imaginary *average student*, who we so often aim to talk to in our mathematics classes, simply didn’t enroll in my course! At the same time, this diversity made me feel more responsible in the following two ways:

- •
The opportunity to introduce the basics of an active mathematical research field to a student who was well on their way to becoming a professional chemist, or a businessperson put me in a position to further encourage the student’s obvious interest in mathematics and to increase the student’s awareness about mathematics as an ever-growing body of human knowledge.

- •
Such a diverse group of students brought to the class a wide range of academic and personal skills, interests, and talents. It was a tempting opportunity for me to create a learning environment where my students would apply those possibly non-mathematics-related proficiencies in the context of Ramsey theory.

Consequently, as part of the course assessment, I asked my students to complete group term projects. I explained to the class that the main purpose of the project was to give each student an opportunity to understand a subset of topics discussed in class in a more substantial way.

This approach was inspired by Kieran Egan (1942–2022), an Irish educational philosopher. In my notes, that I had taken during one of Eagan’s presentations on the topic of “Learning in Depth”

“Learning in Depth” (LiD) is a simple though radical innovation in curriculum and instruction designed to ensure that all students become experts about something during their school years. To the surprise of many, children usually take to the program with great enthusiasm, and within a few months LiD begins to transform their experience as learners.

At the beginning of the semester, I provided a list of possible projects and invited my students to form teams on their own or to send me their preference so that I could group them based on their interests. The list included a wide range of ideas. For example, I invited my students to create a math video related to a topic from the course and post it on an online video sharing platform; or to create an applet or a computer game based on a theorem discussed in the class; or to create a play or a dance or a music piece or a graphic novel inspired by a course topic; or to become a true expert and write a paper about a particular segment of the course, and so on.

My general approach was to give my students as much creative freedom as they wanted.

Each project was evaluated based on three criteria: mathematical content, originality, and complexity. In my opinion, each of the completed projects provided an excellent example of how much talent there is among this generation of students and how mathematics can trigger students’ creativity in sometimes unpredictable ways.

## Term Projects

Over the four times it has been offered, 226 students have taken my Introduction to Ramsey Theory course and completed 60 term projects. These projects have ranged from a video entitled *Ramsey Theory—Introduction* to posting the mathematician Thomas C. Brown’s Wikipedia page to a music theme inspired by van der Waerden’s theorem to a beamer presentation about Roth’s theorem.Footnote^{2}

The diversity of the projects reflected the diversity of students’ academic backgrounds, their levels of mathematical knowledge, and their plans for their future careers. Nevertheless, all students shared their joy of doing mathematics.

There have been several outstanding projects, including an app for finding convex-gons, for among a set of points in general position; or a version of the tic-tac-toe game in a , cube, inspired by the Hales-Jewett theorem;Footnote^{3} or a visual proof of the fact that the Ramsey number equals to 17.

Among my personal favorites, was a project completed by two engineering students who made a “model of a special case of Ramsey’s theorem that an edge 2-coloring of a complete graph of size 6 will always have a monochromatic triangle.” In the students’ words: “To demonstrate Ramsey theorem in the real world, we have incorporated different aspects of engineering in our project.”

In several projects (“become a true expert” and “enhance the course lecture notes”) students demonstrated their capability to learn and do mathematics well above the expected level of a Mathematical Journeys course. For example, members of the group that studied the proof of Roth’s theorem were able to find a very subtle typo in the source that they were using. “What do we do?” they asked. “Let the author know,” I replied. The next class, they were all smiles: “The author replied and thanked us!”

Still, the part that I enjoyed the most was witnessing how mathematics can inspire students’ imagination and creativity and put them in a position to demonstrate their artistic talents and skills, provide them with the space to show their sense for humor, manifest their curiosity about the world around us, and showcase their knowledge of modern technology and popular culture.

The experience with managing my classes through the pandemic also made me much more proficient in using the available technology to document my students’ work. For example, I used PreTeXt,Footnote^{4} an authoring and publishing system, to create an online collection of almost all students’ projects from 2020, 2021, and 2024 and several projects from 2014.

^{4}

See https://pretextbook.org/ and the article by Oscar Levin later in this Early Career section.

Not everything went smoothly. A small number of projects were not completed to my satisfaction. For example, a group of students who studied the chromatic number of the plane,Footnote^{5} including de Gray’s proof that ,

## Ramsey Theory Podcast

In September 2021, ten students chose to create a podcast with interviews of mathematicians whose research interests included Ramsey theory as their term project. I divided them into three groups. During our initial meetings, I provided each group with about a dozen names of mathematicians whose research interests included Ramsey theory. In my follow up message, I set relatively modest expectations:

Do not be too disappointed if you do not get many replies to your invitation. You are reaching out to a group of very busy people who often do not see the value in talking about and promoting what they do to a general audience.

It turned out that I was wrong. About two-thirds of mathematicians that my students initially contacted replied. Most of those replies were positive, ranging from agreeing for an interview to asking for more information about the podcast. After clarifying the nature of the podcast, i.e., that this was an outreach rather than a scientific project and not being able to resolve several scheduling problems, my students ended up conducting 14 interviews with some of the most notable mathematicians of today whose research interests include Ramsey theory.

Students recorded about 18 hours of interviews in total. It turned out that editing this amount of material and creating a podcast with 14 uniformly structured episodes, was a job in itself. I hired one of my former students to do this segment of the podcast project.Footnote^{6}

^{6}

All 14 podcast episodes are available on Spotify (https://open.spotify.com/show/4UrTrYkJc9rFhiOQRoNbm3), Apple Podcasts, and Google Podcasts.

The podcast’s title *Ramsey Theory Podcast: No Strangers at This Party* came from the well-known special case of Ramsey’s theorem that is commonly stated in terms of *friends and strangers*.Footnote^{7}

As one of the students summarized, “The podcast’s listeners will be treated to a deeper understanding of these brilliant, personable, and down-to-earth people we call mathematicians. We hoped to get to know [our guests] on a more personal level to show … how approachable mathematics can be. In the end, we hope that when it comes to mathematics, there will be ‘no strangers at this party.”’

Personally, coordinating this project made me even more proud of my students and our Ramsey theory community. For me, one of the outcomes of the project was that it provided another example that the love for mathematics has the power to unite us across continents, nationalities, ages, genders, and ranks.

## Conclusion

It takes a lot of work, devotion, and mutual trust to create an environment where the process of learning mathematics will be both the inspiration for and the beneficiary of skills and talents that each student brings to their math class.

In my view, it is the instructor’s responsibility to create such an environment whenever possible. Introducing advanced topics to students who are not necessarily planning to become professional mathematicians means investing in future allies and promoters of mathematics among the general public. It also means supporting students’ possible lifelong love for mathematics, regardless of the fact that they might chose a different career path. As one of my engineering students put it:

The podcast project we did was extremely helpful. It made me realize once again the reason for liking math. Talking with great mathematicians inspired me a lot. I realized my passion is not lost. And I am very grateful to you for that chance.

And finally, it may be that the experience in the Ramsey theory course has cracked a portal into the world of mathematical research for one of my former students. This is from a message that I received from a student who was enrolled in my course under the so-called Concurrent Studies program that allows high school students to take university courses:

I am still waiting for a reply from the [Journal] as for my [term-project-related] paper.

Regarding my future studies, I’ve been accepted to [University], and would be interested in continuing to learn more on Ramsey theory there. Would you have any suggestions on professors, courses, or research groups?

## Acknowledgment

The author would like to thank Dr. Angela Gibney and Dr. Krystal Taylor for their guidance and thoughtful comments during the preparation of this article.

## References

[ 1] - Aubrey D. N. J. de Grey,
*The chromatic number of the plane is at least 5*, Geombinatorics**28**(2018), no. 1, 18–31. MR3820926,## Show rawAMSref

`\bib{deGrey}{article}{ author={de Grey, Aubrey D. N. J.}, title={The chromatic number of the plane is at least 5}, journal={Geombinatorics}, volume={28}, date={2018}, number={1}, pages={18--31}, issn={1065-7371}, review={\MR {3820926}}, }`

[ 2] - K. Egan,
*Learning in Depth: A Simple Innovation that Can Transform Schooling*, University of Chicago Press, Chicago, Illinois, 2010. [ 3] - Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer,
*Ramsey theory*, 2nd ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., New York, 1990. A Wiley-Interscience Publication. MR1044995,## Show rawAMSref

`\bib{Graham}{book}{ author={Graham, Ronald L.}, author={Rothschild, Bruce L.}, author={Spencer, Joel H.}, title={Ramsey theory}, series={Wiley-Interscience Series in Discrete Mathematics and Optimization}, edition={2}, note={A Wiley-Interscience Publication}, publisher={John Wiley \& Sons, Inc., New York}, date={1990}, pages={xii+196}, isbn={0-471-50046-1}, review={\MR {1044995}}, }`

[ 4] - V. Jungić,
*An introduction of the problem of finding the chromatic number of the plane, Part I*, Crux Mathematicorum**45**(2020), no. 8, 390–396. [ 5] - V. Jungić,
*An introduction of the problem of finding the chromatic number of the plane, Part II*, Crux Mathematicorum**47**(2021), no. 8, 384–391. [ 6] - Veselin Jungić,
*Basics of Ramsey theory*, CRC Press, Boca Raton, FL, 2023. With a foreword by Thomas C. Brown. MR4781338,## Show rawAMSref

`\bib{Jungic23}{book}{ author={Jungi\'{c}, Veselin}, title={Basics of Ramsey theory}, note={With a foreword by Thomas C. Brown}, publisher={CRC Press, Boca Raton, FL}, date={2023}, pages={xvii+219}, isbn={978-1-032-26037-2}, isbn={978-1-032-26066-2}, isbn={978-1-003-28637-0}, review={\MR {4781338}}, }`

[ 7] - Bruce M. Landman and Aaron Robertson,
*Ramsey theory on the integers*, 2nd ed., Student Mathematical Library, vol. 73, American Mathematical Society, Providence, RI, 2014, DOI 10.1090/stml/073. MR3243507,## Show rawAMSref

`\bib{Landman}{book}{ author={Landman, Bruce M.}, author={Robertson, Aaron}, title={Ramsey theory on the integers}, series={Student Mathematical Library}, volume={73}, edition={2}, publisher={American Mathematical Society, Providence, RI}, date={2014}, pages={xx+384}, isbn={978-0-8218-9867-3}, review={\MR {3243507}}, doi={10.1090/stml/073}, }`

[ 8] - Aaron Robertson,
*Fundamentals of Ramsey theory*, Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, 2021, DOI 10.1201/9780429431418. MR4436191,## Show rawAMSref

`\bib{Robertson}{book}{ author={Robertson, Aaron}, title={Fundamentals of Ramsey theory}, series={Discrete Mathematics and its Applications (Boca Raton)}, publisher={CRC Press, Boca Raton, FL}, date={2021}, pages={xiii+241}, isbn={978-1-138-36433-2}, isbn={978-0-429-43141-8}, review={\MR {4436191}}, doi={10.1201/9780429431418}, }`

[ 9] - Alexander Soifer,
*The mathematical coloring book*:*Mathematics of coloring and the colorful life of its creators*, Springer, New York, 2009. With forewords by Branko Grünbaum, Peter D. Johnson, Jr. and Cecil Rousseau. MR2458293,## Show rawAMSref

`\bib{Soifer}{book}{ author={Soifer, Alexander}, title={The mathematical coloring book}, note={With forewords by Branko Gr\"{u}nbaum, Peter D. Johnson, Jr. and Cecil Rousseau}, subtitle={Mathematics of coloring and the colorful life of its creators}, publisher={Springer, New York}, date={2009}, pages={xxx+607}, isbn={978-0-387-74640-1}, review={\MR {2458293}}, }`

## Credits

Photo of Veselin Jungić is courtesy of Veselin Jungić.