Zero-sum Ramsey theory: Origins, present, and future

Abstract: As for the origins, I will describe the birth of the Erd\H os-Ginzburg-Ziv theorem as I learned it from the late A. Ziv and A. Ginzburg in 2003. A stimulating conversation with V. Milman around 1980 led me to a broader view of Ramsey Theory. I shared some of the ideas with my friends Y. Caro and Y. Roditty. Y. Caro took a slightly different turn and made several significant contributions. In the mid 80's I started my 15-year collaboration with my colleague P. Dierker. In 1989, R. Graham learned about the zero-sum tree conjecture and popularized it. It was solved by Z. F\" uredi and D. Kleitman, and, independently, by A. Schrijver and P. D. Seymour. In 1990 I visited Australia and was introduced to a young student M. Kisin, who made a significant contribution toward the multiplicity conjecture, solved asymptotically by Z. Fuͤredi and D. Kleitman. Another milestone was my joint paper with P. Erdős and H. Lefman, which was the beginning of zero-sum theory on the integers. A few of my Ph.D students and some of my REU students, among them D. Grynkiewicz, made some significant contributions. But I believe that my last Ph.D student, T.D. Luong, paved the way to future research, which I will call vanishing polynomials. Though the abstract describes mainly the history, much of the lecture will be devoted to the present and the future.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2021)

Series comments: This is the nineteenth in a series of annual workshops sponsored by the New York Number Theory Seminar on problems in combinatorial and additive number theory and related parts of mathematics.

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