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

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

Consider functions $f:S\to \coprod_{x\in\mathbb{R}}S_x$ such that $f\circ \pi=\mathrm{id}$ where $\pi:\coprod_{x\in\mathbb{R}}S_x\to S$ is the projection. They are indexed by $2^{\mathbb{R}}$ because each $s\in S$ can go in $2$ subsets. Is there $f$ such that $\lvert f(S)\cap S_x\rvert<\infty$ for each $x\in \mathbb{R}$?