Abstract: A convex geometric hypergraph or cgh is a hypergraph whose vertex set comprises the vertices of a convex polygon in the plane. The extremal problem consists in determining, given an $r$-uniform cgh $F$, the maximum number of edges $\mbox{ex}(n,F)$ in an $r$-uniform cgh on $n$ vertices that does not contain $F$. In the case of graphs, this problem has a rich history with applications to a variety of problems in combinatorial geometry. In this talk, we survey some of the known results for convex geometric graphs, and determine tight bounds for the extremal function for most configurations consisting of a pair of triangles, using a variety of different methods. For instance, we show that the maximum number of triangles that can be formed amongst $n \geq 3$ points in the plane so that any two of the triangles share an interior point is $n(n - 1)(n + 1)/24$ if $n$ is odd and $n(n - 2)(n + 2)/24$ if $n$ is even. Our results generalize old work of Kupitz, Perles, Capoyleas-Pach, Brass, Brass-Karolyi-Valtr and recent work of Aronov-Dujmovic-Morin-Ooms-da Silveira, and answer recent questions of Frankl, Holmsen and Kupavskii.
Joint work with Z. Furedi, T. Jiang, A. Kostochka, D. Mubayi and J. O'Neill.
For Zoom information, please contact Sean at SEnglish (at) illinois (dot) edu.