Complexity of a Single Face in an Arrangement of s-Intersecting Curves
  Consider a face F in an arrangement of n Jordan curves in the plane, no two of which intersect more than s times. We prove that the combinatorial complexity of F is O(\lambda_s(n)), O(\lambda_{s+1}(n)), and O(\lambda_{s+2}(n)), when the curves are bi-infinite, semi-infinite, or bounded, respectively; \lambda_k(n) is the maximum length of a Davenport-Schinzel sequence of order k on an alphabet of n symbols.   Our bounds asymptotically match the known worst-case lower bounds. Our proof settles the still apparently open case of semi-infinite curves. Moreover, it treats the three cases in a fairly uniform fashion. 
Boris Aronov, Dmitriy Drusvyatskiy
cs.CG math.CO