Graduiertenkolleg: Methods for Discrete Structures

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Monday Lecture and Colloquium

Monday, January 24, 2011

Technische Universität Berlin
Fakultät II, Institut für Mathematik
Straße des 17. Juni 136
10623 Berlin

room MA 041

Lecture - 14:15

Frank Lutz - TU Berlin / BTU Cottbus

Discretization of Manifolds

Triangulations, quadrangulations and more general cell decompositions of surfaces and higherdimensional manifolds sometimes surprisingly arise in rather different areas of mathematics. In this lecture, I will give a survey of combinatorial, topological and geometric aspects of discretized manifolds.

Colloquium - 16:00

Dmitry Shabanov - Moskau

Van der Waerden number and colorings of hypergraphs

The van der Waerden number W(n,r) is the smallest integer N such that in any r-coloring of the set {1,2,...,N} there exists a monochromatic arithmetic progression of length n. Famous van der Waerden's theorem states that W(n,r) is finite for all n and r, but does not give any reasonable estimates. This theorem is one of the fundamental results of Ramsey theory. We shall discuss the known bounds for the van der Waerden number. The best upper bounds follow from the density results concerning arithmetic progressions, but the estimating W(n,r) from below finds a very close connection with well-known extremal problems concerning colorings of hypergraphs. We shall present a new asymptotic lower bound for the van der Waerden number obtained by applying the random recoloring method.

Letzte Aktualisierung: 21.01.2011