We study the following family of problems: Given a set of $n$ points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of *forbidden configurations*. As forbidden configurations we consider all 8 ways in which a pair of triangles in such a point set can interact. This leads to 256 extremal Turán-type questions. We give nearly tight (within a $\log n$ factor) bounds for 248 of these questions and show that the remaining 8 questions are all asymptotically equivalent to Stein’s longstanding tripod packing problem.

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Publication

To appear in *The Electronic Journal of Combinatorics*