Monochromatic quotients, products and polynomial sums in the rationals

Let k, a is an element of N and let p1, center dot center dot center dot, pk is an element of Q[n] with zero constant term. We show that for any finite coloring of Q, there are non-zero x, y is an element of Q such that there exists a color which contains the set { } x, yx a ,x + p1(y),center dot center dot center dot,x + pk(y) and there are non-zero v, u is an element of Q such that there exists a color which contains the set { } v, v center dot ua, v + p1(u),center dot center dot center dot , v + pk(u) . (c) 2023 Elsevier B.V. All rights reserved.