OPEN

Let $(A_i)$ be a family of countable sets such that $\lvert A_i\cap A_j\rvert \neq 2$ for all $i\neq j$. Is there some $C$ such that $\cup A_i$ can always be coloured with at most $C$ colours so that no $A_i$ is monochromatic?

A problem of Komjáth. If instead we have $\lvert A_i\cap A_j\rvert \neq 1$ then Komjáth showed that this is possible with at most $\aleph_0$ colours.