Monochromatic sums and products in N

• 成果类型: Article
• 署名作者: Moreira, Joel
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2017.185.3.10
• 发表日期: 2017
• 页码: 1069-1090
• 关键词: large sets; POLYNOMIALS; szemeredi; density; THEOREM; van

摘要

An old question in Ramsey theory asks whether any finite coloring of the natural numbers admits a monochromatic pair {x + y, xy}. 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 x(2) - y(2) = z and x(2) + 2y(2) - 3z(2) = w.