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Section 18.5 Conclusion

This paper presents the new van der Waerden number

\begin{equation*} W(21;2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3)=52. \end{equation*}

This result not only expands the current knowledge of van der Waerden numbers but also demonstrates the effectiveness of computational techniques in tackling these complex combinatorial problems.

The discovery of this new van der Waerden number contributes to the mathematical community's knowledge of these numbers and ongoing advancements in computational power and algorithms promise continued progress in this fascinating area of combinatorics.

\(\textbf{Acknowledgments:}\) The author would like to thank Tanbir Ahmed and Veselin Jungic for making this paper possible. Tanbir Ahmed provided the computational techniques the results are based on, and Veselin Jungic provided an opportunity to work on this problem.