Let $S$ be a set with $|S|=|\mathbb{R}|$. Suppose it has subsets $S_x$ indexed by $x\in \mathbb{R}$ with $|S_x|=|\mathbb{R}|$ for each $x\in \mathbb{R}$. Suppose that

• for any $s\in S$ we have $|\{x\in\mathbb{R}|s\in S_x\}|=2$
• for any $x\neq y\in \mathbb{R}$ we have $|S_x\cap S_y|=3$.

Is there a function $f:S\to \coprod_{x\in\mathbb{R}}S_x$ such that $f\circ \pi=\mathrm{id}$ where $\pi:\coprod_{x\in\mathbb{R}}S_x\to S$ is the projection and such that $|f(S)\cap S_x|<\infty$ for all $x\in \mathbb{R}$?