# Van der Waerden's Theorem

 Title: Van der Waerden's Theorem Authors: Katharina Kreuzer and Manuel Eberl Submission date: 2021-06-22 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 $l$. BibTeX: @article{Van_der_Waerden-AFP, author = {Katharina Kreuzer and Manuel Eberl}, title = {Van der Waerden's Theorem}, journal = {Archive of Formal Proofs}, month = jun, year = 2021, note = {\url{https://isa-afp.org/entries/Van_der_Waerden.html}, Formal proof development}, ISSN = {2150-914x}, } License: BSD License Status: [ok] This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.