Wanting to learn a bit about Ramsey's theory, I read the corresponding article on Wikipedia and stumbled upon this:


"Le théorème de van der Waerden[2] : pour tous entiers c et n, il existe un entier[3] W tel que si l'ensemble {1, 2, … , W} est coloré avec c couleurs, il contient une progression arithmétique monochrome de longueur n."


Fast English translation: for all integers $c$ and $n$, there exist an integer $W$ such that if the set $\{1, 2, 3,\cdots W\}$ is colored with $c$ colors, it contains a monochromatic arithmetic progression of length $n$.


So, coloring an integer $m$ with the color $k$ if and only $\omega(m)\equiv k\pmod c$, one should attain a threshold $W$ such that the set of prime powers lato sensu is monochromatic and contains an arithmetic progression of length $n$ with probability roughly $1/c$. Can one expect the length of the corresponding monochromatic arithmetic progression to be unbounded with probability 1 and from this deduce Green-Tao's theorem?



Is Green-Tao's theorem a consequence of Van der Waerden theorem?
The primes are a density zero subset of the natural numbers. Therefore, van der Waerden's theorem or its great strengthening, Szemerédi's theorem is not directly applicable. In fact the idea of Green and Tao is to "make Szemerédi's theorem work for primes" by proving a "transference theorem" from the natural numbers to the primes.


