Let $G$ be a infinite group. How to show that there is a $A\subseteq G$ infite such that $$\forall x,y,z\in A\;\; \big(xy=z\Leftrightarrow (x=y=z=x^2)\big)$$
I've tried to define the coloring $c:[G]^3\to \{0,1\}$ by $$ c(\{x,y,z\})= \begin{cases} 0&,\text{ if } (xy=z)\Rightarrow(x=y=z=x^2); \text{ or}\\ 1&, \text{ otherwise}. \end{cases} $$ By Ramsey theorem, there is an infinite set $A\subseteq G$ such that $c\restriction [A]^3$ is constant. As I'm working in $[A]^3$, the negation of $x=y=z=x^2$ becomes $z\neq x^2$; but, assuming that, I was not able to reach a contradiction.