We consider problems related to finding short cycles, small cliques, small independent sets,
and small subgraphs in geometric intersection graphs. We obtain a plethora of new results …

A graph drawn in the plane is called k-quasi-planar if it does not contain k pairwise crossing
edges. It has been conjectured for a long time that for every fixed k, the maximum number of …

A string graph is the intersection graph of a collection of continuous arcs in the plane. We
show that any string graph with m edges can be separated into two parts of roughly equal …

An intersection graph of curves in the plane is called a string graph. Matoušek almost
completely settled a conjecture of the authors by showing that every string graph with m …

An outerstring graph is the intersection graph of curves lying inside a disk with one endpoint
on the boundary of the disk. We show that an outerstring graph with $ n $ vertices has …

A topological graph is k-quasi-planar if it does not contain k pairwise crossing edges. A 20-
year-old conjecture asserts that for every fixed k, the maximum number of edges in a k-quasi …

Outerstring graphs are the intersection graphs of curves that lie inside a disk such that each
curve intersects the boundary of the disk. Outerstring graphs are among the most general …

A simple topological graph is k-quasiplanar (k≥ 2) if it contains no k pairwise crossing
edges, and k-planar if no edge is crossed more than k times. In this paper, we explore the …

" Based on a lecture series given by the authors at a satellite meeting of the 2006
International Congress of Mathematicians and on many articles written by them and their …

We show that any set of n points in general position in the plane determines n 1− o (1)
pairwise crossing segments. The best previously known lower bound, Ω (√ n), was proved …