On Monochromatic Solutions of Some Nonlinear Equations in $\mathbb Z/p\mathbb Z$

Let the set of positive integers be colored in an arbitrary way in finitely many colors (a “finite coloring”). Is it true that, in this case, there are $x,y\in\mathbb Z$ such that $x+y$, $xy$, and $x$ have the same color? This well-known problem of the Ramsey theory is still unsolved. In the present paper, we answer this question in the affirmative in the group $\mathbb Z/p\mathbb Z$, where $p$ is a prime, and obtain an even stronger density result.