A Tura´n-type theorem on chords of a convex polygon
Journal of Combinatorial Theory Series B
Journal of Graph Theory
Quasi-Planar Graphs Have a Linear Number of Edges
GD '95 Proceedings of the Symposium on Graph Drawing
Graph Drawings with no k Pairwise Crossing Edges
GD '97 Proceedings of the 5th International Symposium on Graph Drawing
Topological Graphs with No Large Grids
Graphs and Combinatorics
On the maximum number of edges in topological graphs with no four pairwise crossing edges
Proceedings of the twenty-second annual symposium on Computational geometry
Note: On the maximum number of edges in quasi-planar graphs
Journal of Combinatorial Theory Series A
Crossing Stars in Topological Graphs
SIAM Journal on Discrete Mathematics
Coloring kk-free intersection graphs of geometric objects in the plane
Proceedings of the twenty-fourth annual symposium on Computational geometry
On grids in topological graphs
Proceedings of the twenty-fifth annual symposium on Computational geometry
Improved bounds and new techniques for Davenport--Schinzel sequences and their generalizations
Journal of the ACM (JACM)
Generalized Davenport-Schinzel sequences and their 0-1 matrix counterparts
Journal of Combinatorial Theory Series A
On the structure and composition of forbidden sequences, with geometric applications
Proceedings of the twenty-seventh annual symposium on Computational geometry
h-quasi planar drawings of bounded treewidth graphs in linear area
WG'12 Proceedings of the 38th international conference on Graph-Theoretic Concepts in Computer Science
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A topological graph is k-quasi-planar if it does not contain k pairwise crossing edges. A topological graph is simple if every pair of its edges intersect at most once (either at a vertex or at their intersection). In 1996, Pach, Shahrokhi, and Szegedy [16] showed that every n-vertex simple k-quasi-planar graph contains at most $O\left(n(\log n)^{2k-4}\right)$ edges. This upper bound was recently improved (for large k) by Fox and Pach [8] to n(logn)O(logk). In this note, we show that all such graphs contain at most $(n\log^2n )2^{\alpha^{c_k}(n)}$ edges, where α(n) denotes the inverse Ackermann function and ck is a constant that depends only on k.