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An arrangment of $n$ circles is simple if any two circles are disjoint or intersect in exactly two points. The arrangment is intersecting if every pair of circles is intersecting.

A triangle is an arrangement of circles is a face bouned by three circes. A digon (aka lens) is a face bounded by two circles. Grünbaum made the following conjecture about pseudocircles, but it has been disproved by Felsner and Scheucher. It may still be true for circles though:

Is it true that every simple intersecting digon-free arrangement of $n$ circles contains at least $2n-4$ triangles?

For the original problem (about pseudocircles), Felsner and Scheucher give examples of simple intersecting digon-free arrangements of $n$ circles having only $16n/11 + o(n)$ triangles. A lower bound of $4n/3$ due to Hershberger and Snoeyink is probably the right answer:

Do there exist simple intersecting digon-free arrangments of $n$ pseudocircles with only $4n/3+o(n)$ triangles for infinitely many values of $n$?