A convex geometric hypergraph or cgh consists of a family of subsets of a strictly convex set of points in the plane. There are eight pairwise nonisomorphic cgh's consisting of two distinct triangles. These were studied at length by Braß [6] (2004) and by Aronov, Dujmović, Morin, Ooms, and da Silveira [2] (2019). We determine the extremal functions exactly for seven of the eight configurations.

The above results are about cyclically ordered hypergraphs. We extend some of them for triangle systems with vertices from a non-convex set. We also solve problems posed by P. Frankl, Holmsen and Kupavskii [15] (2020), 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.

凸几何超图或cgh 由平面中一组严格凸点的子集组成。有八个成对的非同构 cgh，由两个不同的三角形组成。Braß [6] (2004) 和 Aronov、Dujmović、Morin、Ooms 和 da Silveira [2] (2019) 对这些进行了详细研究。我们准确地确定了八种配置中的七种的极值函数。

上述结果是关于循环有序超图的。我们将其中一些扩展用于具有非凸集合顶点的三角形系统。我们还解决了 P. Frankl、Holmsen 和 Kupavskii [15] (2020) 提出的问题，特别是，我们确定了顶点来自平面中的一组n个点的三角形相交族的确切最大尺寸。