Holes in Convex and Simple Drawings
<|reference_start|>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\H{o}s--Szekeres theorem is known and empty triangles have been investigated. We introduce a notion of $k$-holes for simple drawings and study their existence with respect to the convexity hierarchy. 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 will be the structural investigation of pseudolinear subdrawings in convex~drawings.<|reference_end|>
@article{bergold2024holes,
    title={Holes in Convex and Simple Drawings},
    author={Helena Bergold, Joachim Orthaber, Manfred Scheucher, Felix Schr"oder},
    journal={arXiv preprint arXiv:2409.01723},
    year={2024},
    archivePrefix={arXiv},
    eprint={2409.01723},
    primaryClass={cs.CG cs.DM math.GT}
}