On levels in arrangements of lines, segments, planes, and triangles

We consider the problem of bounding the complexity of the k-th level in an arrangement of n curves or surfaces, a problem dual to, and extending, the well-known k-set problem. (a) We review and simplify some old proofs in new disguise and give new proofs of the bound O(n√k+1) for the complexity of the k-th level in an arrangement of n lines. (b) We derive an improved version of Lovasz Lemma in any dimension, and use it to prove a new bound, O(n²k^2/3), on the complexity of the k-th level in an arrangement of n planes in R³, or on the number of k-sets in a set of n points in three dimensions. (c) We show that the complexity of any single level in an arrangement of n line segments in the plane is O(n^3/2), and that the complexity of any single level in an arrangement of n triangles in 3-space is O(n^17/6).