Problem 602

Erdős Problem #602: declared status 'open'. Formalized: yes. Let $(A_i)$ be a family of sets with $\lvert A_i\rvert=\aleph_0$ for all $i$, such that for any $i\neq j$ we have $\lvert A_i\cap A_j\rvert$ finite and $\neq 1$. Is there a $2$-colouring of $\cup A_i$ such that no $A_i$ is monochromatic? Prize: no. Tags: combinatorics, set theory.