Will any new Van der Waerden number be known by the end of 2030?
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The Van der Waerden number W(r,k) is equal to the smallest N such that, if the numbers 1 through N are colored with r colors, there is always an arithmetic progression of length at least k with the same color.
Trivially, W(1,k) = k, W(r,1)=1, and W(r,2) = r+1 for all k,r. However, all other values are difficult to compute, and only 7 are known to date:
W(2,3) = 9
W(2,4) = 35
W(2,5) = 178
W(2,6) = 1132
W(3,3) = 27
W(3,4) = 293
W(4,3) = 76
Will any other Van der Waerdin number be known by the end of 2030? This will be based on whenever the discovery is published, and the closure date is a year later than the deadline to ensure that I don't miss it if it is published right before 2030 ends.
This question is managed and resolved by Manifold.
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