In this paper we prove that for any finite coloring of N there are lambda,rho in N such that infinitely many pairs (x,y),(u,v) in N^2 satisfy the sets { lambda x, lambda y, x y, lambda(x+y)} and { u+rho, v+rho, u v+rho, u+v} being monochromatic. Using related arguments we also give two different proofs of a special case of the Milliken--Taylor theorem.
Categories: math.CO math.DS math.NT