  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.
