Some Lower Bounds for Three Color Van Der Waerden Numbers

Let k ≥ 2, a k-termarithmetic progression is a sequence of positive integers {a₁, a₂, ..., a_k} such that there is a constant positive integer d with the property that a _i+1 – a_i = d for i = 1,2, ... , k – 1. The van der Waerden number W (k ₁, k ₂, ... , k _r) is the smallest integer ω such that every r-coloring of {1, 2, ... , ω} contains a monochromatic k_i -term arithmetic progression with color i for some i. In this paper, some computational techniques such as randomized search, forward check-ing, are used to find lower bounds for van der Waerden numbers. As a result, five lower bounds are obtained: W (3, 4, 6) ≥ 178, W (3, 4, 7) ≥ 230, W (3, 4, 8) ≥ 291, W (3, 4, 9) ≥ 366, W (3, 4, 10) ≥ 388.