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Lecture 4: Geometric Graphs | Einstein Institute of Mathematics

Lecture 4: Geometric Graphs

Date: 
Fri, 21/05/200412:20-13:20
Location: 
Hall 2
Lecturer: 
Prof. János Pach (Courant Institute, N.Y.U.)

According to Euler's formula, every planar graph with n vertices has at most \(O(n)\) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and
recent results of this kind, we prove that every graph of $n$ vertices, which can be drawn in the plane
without three pairwise crossing edges, has at most $O(n)$ edges. What happens if the forbidden pattern consists of four pairwise crossing edges?
These questions can be regarded as dual counterparts of some old problems of Erdos, Kupitz, Perles, and others. Why are they interesting?