Partition regularity and multiplicatively syndetic sets
Tom 196 / 2020
Acta Arithmetica 196 (2020), 109-138
MSC: Primary 11B30; Secondary 05D10.
DOI: 10.4064/aa190421-11-3
Opublikowany online: 19 June 2020
Streszczenie
We show how multiplicatively syndetic sets can be used in the study of partition regularity of dilation invariant systems of polynomial equations. In particular, we prove that a dilation invariant system of polynomial equations is partition regular if and only if it has a solution inside every multiplicatively syndetic set. We also adapt the methods of Green–Tao and Chow–Lindqvist–Prendiville to develop a syndetic version of Roth’s density increment strategy. This argument is then used to obtain bounds on the Rado numbers of configurations of the form $\{x,d,x+d,x+2d\}$.
