Van der Waerden's Theorem

Katharina Kreuzer 🌐 and Manuel Eberl 🌐

June 22, 2021

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.


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 $l$.


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Session Van_der_Waerden