Van Der Waerden's Theorem: Exposition and Generalizations

A Pearl of Number Theory on the Shore of Combinatorics

We shall discuss some history, state several generalizations and mention few applications of the theorem. As we shall see, the theorem of Hales and Jewett, revealing the combinatorial nature of van der Waerden's theorem would claim that this ‘pearl of number theory’ belonged to the ancient shore of Combinatorics. Yet, there remains very difficult unsolved problems arising out of this theme and ...

A constructive topological proof of van der Waerden's theorem

On van der Waerden's Theorem on Arithmetic Progressions

We give a simplified version of a proof of van der Waerden’s theorem that if a sufficiently long interval of integers is partitioned into a specified number of parts, one will contain an arithmetic progression of given length.

Van der Waerden's Theorem and Avoidability in Words

Independently, Pirillo and Varricchio, Halbeisen and Hungerbühler and Freedman considered the following problem, open since 1992: Does there exist an infinite word w over a finite subset of Z such that w contains no two consecutive blocks of the same length and sum? We consider some variations on this problem in the light of van der Waerden’s theorem on arithmetic progressions.

On Applications of Van Der Waerden's Theorem

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