Generalized sum-product phenomena and a related coloring problem

Graph Theory Seminar
Tuesday, October 20, 2020 - 3:45pm for 1 hour (actually 50 minutes)
Location For password, please email Anton Bernshteyn (bahtoh ~at~
Yifan Jing – University of Illinois at Urbana-Champaign – yifanjing17@gmail.com
Anton Bernshteyn

In the first part of the talk, I will show that for two bivariate polynomials $P(x,y)$ and $Q(x,y)$ with coefficients in fields with char 0 to simultaneously exhibit small expansion, they must exploit the underlying additive or multiplicative structure of the field in nearly identical fashion. This in particular generalizes the main result of Shen and yields an Elekes-Ronyai type structural result for symmetric nonexpanders, resolving a question mentioned by de Zeeuw (Joint with S. Roy and C-M. Tran). In the second part of the talk, I will show how this sum-product phenomena helps us avoid color-isomorphic even cycles in proper edge colorings of complete graphs (Joint with G. Ge, Z. Xu, and T. Zhang).