An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair fx+y; xyg. We answer this question affirmatively in a strong sense by exhibiting a large new class of nonlinear patterns that can be found in a single cell of any finite partition of N. Our proof involves a correspondence principle that transfers the problem into the language of topological dynamics. As a corollary of our main theorem we obtain partition regularity for new types of equations, such as x2 - y2 = z and x2 + 2y2 - 3z2 = w.
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty