A Note on Erdős Problem #602

We prove that for any family of sets \(\{A_i\}i∈ I\) such that each \(A_i\) is infinite and all pairwise intersections are finite, there exists a \(2\)-colouring of \(i∈ I A_i\) with no monochromatic \(A_i\).This answers Erdős Problem #602 in the affirmative and shows that the original hypothesis \( A_i∩ A_j ≠ 1\) is superfluous.Two compactness proofs are given: one using Tychonoff's theorem on product spaces, and one using the finite intersection property (propositional compactness).