Extremal problems for pairs of triangles in a convex polygon

Zoltán Füredi, Dhruv Mubayi, Jason O'Neill, Jacques Verstraëte

Published 2020-10-21Version 1

A convex geometric hypergraph or cgh consists of a family of subsets of a strictly convex set of points in the plane. The study of cghs is motivated by problems in combinatorial geometry, and was studied at length by Bra{\ss} and by Aronov, Dujmovi\'c, Morin, Ooms and da Silveira. In this paper, we determine the extremal functions for five of the eight configurations of two triangles exactly and another one asymptotically. We give conjectures for two of the three remaining configurations. Our main results solve problems posed by Frankl, Holmsen and Kupavskii on intersecting triangles in a cgh. In particular, we determine the exact maximum size of an intersecting family of triangles whose vertices come from a set of $n$ points in the plane.