Aaron Robertson (mathematician)

Aaron Robertson was born in Torrance, California. At the age of 5, Robertson and his parents moved to Midland, Michigan. He spent most of his early life in Michigan and subsequently attended the University of Michigan in 1989 where he studied actuarial science as an undergraduate. Robertson graduated with the highest distinction (top 3% of graduating class) in 1993 but was unsure about his career as he was not content with actuarial science. His differential equations teacher suggested to apply for graduate school at the University of Michigan, but Robertson ultimately chose to go to Temple University in Philadelphia, Pennsylvania. In Philadelphia, Robertson met his wife, and they eventually got married there.

One of Robertson's lectures at Temple University was taught by Doron Zeilberger to which he became inspired by his exciting and distinctive talks. After passing the qualifying exam, Zeilberger asked Robertson if he could be his advisee and Robertson agreed. Although Zeilberger's primary focus was not Ramsey Theory, Robertson surprisingly received a Ramsey Theory problem from Zeilberger for his dissertation. In 1999, Robertson received his Ph.D. with his thesis titled Some New Results in Ramsey Theory[2] and immediately after went to Colgate University located in Hamilton, New York. At Colgate University, Robertson became an assistant professor of mathematics. When asked about his experience, he responded: "As an assistant professor it is probably the hardest time for any academic. You are learning to teach courses and trying to get research out, and if you start a family as I did, you have young kids, so you are not sleeping."

After being an assistant professor for six years, in 2005, Robertson received academic tenure and became an associate professor. For the first four years of being an associate professor, Robertson was required to contribute to the university to satisfy Colgate's service requirements. At the moment, he is a full professor at Colgate University and rotates between 8 courses.

Robertson's work in mathematics since 1998 has consisted predominantly of Ramsey Theory related topics.

One of Robertson's most cited contributions is his dissertation with co-author Doron Zeilberger, where they prove that "the minimum number (asymptotically) of monochromatic Schur Triples that a 2-colouring of ${\displaystyle [1,n]}$ can have ${\textstyle n^{2}/22+O(n)}$".[2] After the completion of his dissertation, Robertson then worked with 3-term arithmetic progressions where he found the best-known values that were close to each other and titled this piece New Lower Bounds for Some Multicolored Ramsey Numbers.[3]

Another piece of research that has been cited considerably is a paper co-authored with Doron Zeilberger and Herbert Wilf titled Permutation Patterns and Continued Fractions.[4] In the paper, they "find a generating function for the number of (132)-avoiding permutations that have a given number of (123) patterns"[4] with the result being "in the form of a continued fraction".[4] Robertson's contribution to this specific paper includes discussion on permutations that avoid a certain pattern but contain others.

A notable paper Robertson wrote titled A Probalistic Threshold For Monochromatic Arithmetic Progressions[5] explores the function ${\displaystyle f_{r}(k)={\sqrt {k}}\cdot r^{\tfrac {k}{2}}}$ (where ${\displaystyle r\geq 2}$ is fixed) and the r-colourings of ${\displaystyle [1,n_{k}]=\{1,2,...,n_{k}\}}$. Robertson analyzes the threshold function for ${\displaystyle k}$-term arithmeric progressions and improves the bounds found previously.

In the last decade, Robertson co-authored with Bruce M. Landman to publish the second edition of Ramsey Theory on the Integers.[6] The book introduced new topics such as rainbow Ramsey theory, an “inequality” version of Schur's theorem, monochromatic solutions of recurrence relations, Ramsey results involving both sums and products, monochromatic sets avoiding certain differences, Ramsey properties for polynomial progressions, generalizations of the Erdős-Ginzberg-Ziv theorem, and the number of arithmetic progressions under arbitrary colourings.

More recently, in 2021, Robertson has published the book titled Fundamentals of Ramsey Theory.[7] Robertson's goal in writing this book was to "help give an overview of Ramsey theory from several points of view, adding intuition and detailed proofs as we go, while being, hopefully, a bit gentler than most of the other books on Ramsey theory".[7] Throughout the book, Robertson discusses several theorems including Ramsey's Theorem, Van der Waerden's Theorem, Rado's Theorem, and Hales-Jewett Theorem.

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• Archive of papers published by Aaron Robertson from ResearchGate