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.
- Pub Date:
- June 2017
- Mathematics - Combinatorics;
- Computer Science - Computational Geometry;
- Computer Science - Discrete Mathematics
- 25 pages, 14 figures, 16 graphics. This version corrects one theorem statement from the original version