Department of Computer Science | Institute of Theoretical Computer Science | CADMO

Prof. Emo Welzl and Prof. Bernd Gärtner

**
Ramsey-type questions in geometry
**

**
Janos Pach, City College New York and Hungarian Academy of Sciences**

A graph is called *H**-free* if it does not contain *H*as an induced subgraph. Erdos and Hajnal raised the following
question. Is it true that for every graph *H* there exists
with the property that any *H*-free graph
with *n* vertices contains either a complete or an empty subgraph
with
vertices? We answer this question in
the affirmative for some special classes of graphs, including
graphs defined by geometric means. In particular, we show that
any family
of segments in general position in the
plane has two disjoint subfamilies, each containing at least
a fixed constant fraction of the members of ,
such
that either every member of the first subfamily crosses every
member of the second, or no member of the first subfamily
crosses any member of the second. This is an unpublished result
of J. Solymosi and the speaker.

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