Intersection Patterns of Planar Sets

Let A= { A₁, … , A_n} be a family of sets in the plane. For 0 ≤ i< n, denote by f_i the number of subsets σ of { 1 , … , n} of cardinality i+ 1 that satisfy ⋂ _i∈σA_i≠ ∅. Let k≥ 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≤ cf_k-1 for some positive constant c depending only on k. Similarly, let b≥ 2 , k> 2 b 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.