Extremal problems for pairs of triangles in a convex polygon Zolt´ an F¨ uredi * Dhruv Mubayi † Jason O’Neill ‡ Jacques Verstra¨ ete ‡ October 22, 2020 Abstract 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 Braß 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. 1 Introduction A convex geometric hypergraph or cgh is a family of subsets of a set of points in strictly convex position in the plane – we assume these points, denoted by Ω n = {v 0 ,v 1 ,...,v n−1 }, are the vertices of some regular n-gon with the clockwise cylic ordering v 0 <v 1 < ··· <v n−1 <v 0 . For an r-uniform cgh F – an r-cgh for short – let the extremal function ex  (n, F ) denote the maximum number of edges in an r-uniform cgh on n points that does not contain F – an F -free cgh. For the rich history of ordered and convex geometric graph problems and their applications, see Tardos [27] and Pach [21, 22] and for cghs, see Braß [3]. In this paper, we concentrated on intersection patterns of pairs of triangles. For r = 3, there are eight configurations of two triangles (we refer interchangeably to triangles and triples or edges when r = 3), depicted below: * Alfr´ ed R´ enyi Institute of Mathematics, Hungarian Academy of Sciences, P.O. Box 127, Budapest, Hungary, H-1364. Research was supported in part by NKFIH grant KH130371 and NKFI–133819. E-mail: z-furedi@illinois.edu † Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago. Research supported by NSF award DMS-1952767. E-mail: mubayi@uic.edu ‡ Department of Mathematics, University of California, San Diego. Research supported by NSF award DMS-1800332. E-mail: jmoneill@ucsd.edu and jacques@ucsd.edu 1 arXiv:2010.11100v1 [math.CO] 21 Oct 2020