1) Message boards : Projects : New Project: Van Der Waerden Numbers (Message 61095) Posted 20 Mar 2015 by Hunter Post: This project is now launched with URL: 123numbers.org.

2) Message boards : Projects : New Project: Van Der Waerden Numbers (Message 60764) Posted 9 Mar 2015 by Hunter Post: Van Der Waerden Numbers (vdwnumbers.org) is a research project that aims to find better lower bounds for these numbers. The sequence of colors BRRBBRRB (where B is blue and R is red) does not have an evenly spaced subsequence of length 3 that are the same color. However, if you add a B to the end, you get BRRBBRRBB, which has the same color blue in positions 1,5, and 9 which are evenly spaced 4 apart. If you add an R to the end, you get BRRBBRRBR, which has R at position 3, 6, and 9. In fact, with only two colors, there is no sequence of length 9 of Bs and Rs that does not have a subsequence of 3 evenly spaced of the same color. Van der Waerden's Theorem states that for any number of colors r and length k, a long enough sequence always has an evenly spaced subsequence of length k of the same color. The smallest length guaranteed to have an evenly spaced subsequence is called the Van Der Waerden Number and is written W(k,r). For example, W(3,2)=9. Only 7 of these numbers are known. This project is to find better lower bounds for Van Der Waerden Numbers by finding sequences like BRRBBRRB using large prime numbers with special properties.

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