Abstract
Let A = {A(1) , ... , A(n)} be a family of sets in the plane. For 0 = 2 be an integer.
We prove that if each k-wise and (k+1)-wise intersection of sets from A is empty, or a single point, or both open and path-connected, then f(k+1) = 0 implies f(k) = 2, k > 2b be integers. We prove that if each k-wise or (k+1)-wise intersection of sets from A has at most b path-connected components, which all are open, then f(k+1) = 0 implies f(k) <= cf(k-1) for some positive constant c depending only on b and k.
These results also extend to two-dimensional compact surfaces.