Erdős Problem #602

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?

#602: [Er87]

combinatorics | set theory

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A problem of Komjáth. The existence of such a $2$-colouring is sometimes known as Property B.

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