New Lower Bounds for van der Waerden Numbers Using Distributed Computing

This project found improved lower bounds for van der Waerden numbers. The number $W(k,r)$ is the length of the shortest sequence of colors with $r$ different colors that guarantees an evenly-spaced subsequence (or arithmetic progression) of the same color of length $k$. We applied existing techniques: Rabung's using primitive roots of prime numbers and Herwig et al's Cyclic Zipper Method as improved by Rabung and Lotts. We used much more computing power, 2 teraflops, using distributed computing and larger prime numbers, through over 500 million, compared to 10 million by Rabung and Lotts. We used $r$ up to 10 colors and $k$ up to length 25, compared to 6 colors and length 12 in previous work. Our lower bounds on $W(k,r)$ grow roughly geometrically with a ratio converging down towards $r$ for $r$ equals 2, 3, and 4. We conjecture that the exact numbers $W(k, r)$ grow at the rate $r$ for large $k$.