A variant of the proof of Van der Waerden's theorem by Furstenberg

Van der Waerdens construction of a splitting eld

In his classic book, Modern Algebra, van der Waerden gave a procedure for factoring polynomials over a …nite-dimensional, separable, simple extension …eld. I believe that there is a nonconstructive component to his proof, and I will indicate where it comes in and why. Although I’m sure that this component could be avoided while staying within the framework that he set down, it is simpler to get...

A Dynamical Proof of Van Der Waerden’s Theorem, Second Approach

In this lecture, the second approach of Van Der Waerden’s theorem will be presented. It is based on Birkhoff’s theorem which will also be proved. The essence of Van Der Waerden’s theorem is that if the set of natural numbers N is partitioned in some way into finitely many classes in any way whatever, then one of these classes necessarily contains arbitrarily long arithmetic progressions.

A constructive topological proof of van der Waerden's theorem

On a Variant of Van Der Waerden’s Theorem

Given positive integers n and k, a k-term quasi-progression of diameter n is a sequence (x1, x2, ..., xk) such that d ≤ xj+1−xj ≤ d+n, 1 ≤ j ≤ k−1, for some positive integer d. Thus an arithmetic progression is a quasi-progression of diameter 0. Let Qn(k) denote the least integer for which every coloring of {1, 2, ..., Qn(k)} yields a monochromatic k-term quasi-progression of diameter n. We obt...

A van der Waerden Variant

The classical van der Waerden Theorem says that for every every finite set S of natural numbers and every k-coloring of the natural numbers, there is a monochromatic set of the form aS+b for some a > 0 and b ≥ 0. I.e., monochromatism is obtained by a dilation followed by a translation. We investigate the effect of reversing the order of dilation and translation. S has the variant van der Waerde...