On grids in topological graphs

Eyal Ackerman, Jacob Fox, János Pach, Andrew Suk

Research output: Contribution to journalArticlepeer-review


A topological graph G is a graph drawn in the plane with vertices represented by points and edges represented by continuous arcs connecting the vertices. If every edge is drawn as a straight-line segment, then G is called a geometric graph. A k-grid in a topological graph is a pair of subsets of the edge set, each of size k, such that every edge in one subset crosses every edge in the other subset. It is known that every n-vertex topological graph with no k-grid has Ok(n) edges. We conjecture that the number of edges of every n-vertex topological graph with no k-grid such that all of its 2k edges have distinct endpoints is Ok(n). This conjecture is shown to be true apart from an iterated logarithmic factor log.

Original languageEnglish
Pages (from-to)710-723
Number of pages14
JournalComputational Geometry: Theory and Applications
Issue number7
StatePublished - Aug 2014


  • Geometric graphs
  • Topological graphs
  • Turán-type problems

ASJC Scopus subject areas

  • Computer Science Applications
  • Geometry and Topology
  • Control and Optimization
  • Computational Theory and Mathematics
  • Computational Mathematics


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