### Abstract

This article formalises the proof of Van der Waerden's Theorem
from Ramsey theory. Van der Waerden's Theorem states that for
integers $k$ and $l$ there exists a number $N$ which guarantees that
if an integer interval of length at least $N$ is coloured with $k$
colours, there will always be an arithmetic progression of length $l$
of the same colour in said interval. The proof goes along the lines of
\cite{Swan}. The smallest number $N_{k,l}$ fulfilling Van der
Waerden's Theorem is then called the Van der Waerden Number.
Finding the Van der Waerden Number is still an open problem for most
values of $k$ and $l$.