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Topic: Courant Geometry Seminar
Replies: 8   Last Post: Jan 25, 2005 9:53 PM

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Ricky Pollack

Posts: 18
Registered: 12/3/04
Courant Geometry Seminar: Janos Pach
Posted: Jan 25, 2005 9:53 PM
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Geometry Seminar
Tuesday October 12 in room 613 WWH at 6:00 P.M

Some Applications of Graph Theory to Geometry

Janos Pach
City College-CUNY and CIMS

Since the beginnings of extremal graph theory (roughly 50 years) ago,
the strength of its new methods has always been best demonstrated by
geometric applications. After giving a brief survey of the new
techniques, we establish some new results, including the following
theorem of Erd"os, Makai and myself. Let $k$ be fixed and $n$ tend to
infinity. Then for any set of $n$ points in the plane with minimal
distance at least one, and for any $t_1,t_2,...,t_k$, the number of
point pairs whose distance is between $t_i$ and $t_i + o(\sqrt{n})$ for
some $i$ is at most $\frac{n^2}{2}(1-\frac{1}{k+1}+o(1))$.

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