2016/05/05 by Moreira, Joel
#Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1605.01469
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 non-linear patterns which can be found in a single cell of any finite partition of ℕ. Our proof involves a correspondence principle which 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.