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Consider a face F in an arrangement of n Jordan curves in the plane, no twoof which intersect more than s times. We prove that the combinatorialcomplexity of F is O(\lambda_s(n)), O(\lambda_{s+1}(n)), andO(\lambda_{s+2}(n)), when the curves are bi-infinite, semi-infinite, orbounded, respectively; \lambda_k(n) is the maximum length of aDavenport-Schinzel sequence of order k on an alphabet of n symbols.  Our bounds asymptotically match the known worst-case lower bounds. Our proofsettles the still apparently open case of semi-infinite curves. Moreover, ittreats the three cases in a fairly uniform fashion.

Complexity of a Single Face in an Arrangement of s-Intersecting Curves