Improving the Use of Cyclic Zippers in Finding Lower Bounds for van der Waerden Numbers

For integers k and l, each greater than 1, suppose that p is a prime with p ≡ 1 (mod k) and that the kth-power classes mod p induce a coloring of the integer segment [0, p− 1] that admits no monochromatic occurrence of l consecutive members of an arithmetic progression. Such a coloring can lead to a coloring of [0, (l − 1)p] that is similarly free of monochromatic l-progressions, and, hence, can give directly a lower bound for the van der Waerden number W (k, l). P. R. Herwig, M. J. H. Heule, P. M. van Lambalgen, and H. van Maaren have devised a technique for splitting and “zipping” such a coloring of [0, p−1] to yield a coloring of [0, 2p−1] which, for even values of k, is sometimes extendable to a coloring of [0, 2(l − 1)p] where both new colorings still admit no monochromatic l-progressions. Here we derive a fast procedure for checking whether such a zipped coloring remains free of monochromatic l-progressions, effectively reducing a quadratic-time check to a linear-time check. Using this procedure we find some new lower bounds for van der Waerden numbers.