Problem Statement
Let $(A_i)$ be a family of countably infinite sets such that $\lvert A_i\cap A_j\rvert \neq 2$ for all $i\neq j$. Find the smallest cardinal $C$ such that $\cup A_i$ can always be coloured with at most $C$ colours so that no $A_i$ is monochromatic.
Categories:
Combinatorics Set Theory
Progress
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.Source: erdosproblems.com/603 | Last verified: January 15, 2026