Gons and holes in point sets have been extensively studied in the literature. For simple drawings of the complete graph a generalization of the Erdős--Szekeres theorem is known and empty triangles have been investigated. We introduce a notion of $k$-holes for simple drawings and survey generalizations thereof, like empty $k$-cycles. We present a family of simple drawings without $4$-holes and prove a generalization of Gerken's empty hexagon theorem for convex drawings. A crucial intermediate step is the structural investigation of pseudolinear subdrawings in convex drawings. With respect to empty $k$-cycles, we show the existence of empty $4$-cycles in every simple drawing of $K_n$ and give a construction that admits only $Θ(n^2)$ of them.

翻译：点集中的凸多边形与空洞在文献中已被广泛研究。对于完全图的简单绘制，已有Erdős--Szekeres定理的推广形式，且空三角形问题已得到探讨。我们为简单绘制引入了$k$-空洞的概念，并综述了其推广形式，如空$k$-环。我们提出了一族不含$4$-空洞的简单绘制，并证明了凸绘制中Gerken空六边形定理的推广形式。一个关键的中间步骤是对凸绘制中伪线性子绘制的结构研究。关于空$k$-环，我们证明了在$K_n$的任意简单绘制中均存在空$4$-环，并给出了一种仅包含$Θ(n^2)$个空$4$-环的构造。