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(λs(n)), O(λs+1(n)), and O(λs+2(n)), when the curves are bi-infinite, semi-infinite, or bounded, respectively; λk(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.