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Gallai-van der Waerden Number

The Gallai-van der Waerden number GW(r;3,3) is the least integer N>=r such that, for every n>=N, every exact r-coloring of {1,...,n} contains a monochromatic or rainbow three-term arithmetic progression. An exact coloring uses all r colors. Monochromatic means that all three colors are equal, and rainbow means that they are distinct (Si 2026). The requirement concerns every interval beyond the threshold, not just {1,...,N}.

Si (2026) determined

GW(3;3,3)=GW(4;3,3)=9

and GW(r;3,3)=r for every r>=5. The proof combines finite exhaustive checks with a theoretical argument for r>=9. These numbers differ from the ordinary van der Waerden numbers, which force only monochromatic arithmetic progressions.

See also

Arithmetic Progression, Ramsey Theory, van der Waerden Number

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References

Si, Y. "Complete Determination of the Gallai-van der Waerden Number GW(r;3,3)." 22 Oct 2026. https://arxiv.org/abs/2610.30711.

Cite this as:

Weisstein, Eric W. "Gallai-van der Waerden Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Gallai-vanderWaerdenNumber.html

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