Edit 2: solved!
In this post, "we" proved that exist infinite $R_1$, such that $f$ is constant on elements of the form $\{n_1,r\}$ where $r\in R_1$. By the same considerations we can show that if we fix $n_2\in R_1$, there is exist infinite $R_2\subseteq R_1-\{n_2\}$ such that $f$ is constant on sets of the form $\{n_2,r\}$ where $r\in R_2$. Now saying that $f(\{\{n_1,r\}:r\in R_1\})=t_1$ and $f(\{\{n_2,r\}:r\in R_2\})=t_2$. I need to show via induction that we can find a sequence of naturals $n_1,n_2,n_3,...$, and a sequence of $t_1,t_2,t_3,...$ where $t_n\in\{1,2\}$ for all $n$, and if $i
Thank you!