Erdős Problem #602
Property B

Does every almost-disjoint family of countably infinite sets whose pairwise intersections all have size ≠ 1 have Property B? Formally: let `α` be any type, let `(A_i)_{i ∈ I}` be a family of countably infinite subsets of `α` such that for all `i ≠ j`, the intersection `A_i ∩ A_j` is finite and `|A_i ∩ A_j| ≠ 1`. Does there exist a 2-colouring `f : α → Fin 2` such that no `A_i` is monochromatic? This is an open question about Property B for almost-disjoint families with a forbidden intersection size of 1. **Note:** This generalises the formulation in which the ground set is `ℕ`. Since every countably infinite set is in bijection with `ℕ`, the two formulations are equivalent, but working over an arbitrary ground type makes the statement apply immediately to, e.g., almost-disjoint families of countable subsets of an uncountable space.