Holes in Convex and Simple Drawings

Gons and holes in point sets have been extensively studied in the literature. For simple drawings of the complete graph a generalization of the Erd{o}s--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 Kₙ and give a construction that admits only Θ(n²) of them.