On the maximum number of edges in topological graphs with no four pairwise crossing edges

Graph Drawings with no k Pairwise Crossing Edges

GD '97 Proceedings of the 5th International Symposium on Graph Drawing

Note: On the maximum number of edges in quasi-planar graphs

Journal of Combinatorial Theory Series A
Coloring kk-free intersection graphs of geometric objects in the plane

Proceedings of the twenty-fourth annual symposium on Computational geometry
k-Quasi-planar graphs

GD'11 Proceedings of the 19th international conference on Graph Drawing
String graphs and incomparability graphs

Proceedings of the twenty-eighth annual symposium on Computational geometry

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We show that the maximum number of edges in a topological graph on n vertices and with no four pairwise crossing edges is O(n).