A note on the mixed van der Waerden number - Sim, Kai An (Korean Mathematical Society, 2021)
Let r >= 2, and let k(i) >= 2 for 1 <= i <= r. Mixed van der Waerden's theorem states that there exists a least positive integer w = w(k(1), k(2), k(3), ..., k(r); r) such that for any n >= w, every r-colouring of 1, n] admits a k(i)-term arithmetic progression with colour i for some i is an element of 1, r]. For k >= 3 and r >= 2, the mixed van der Waerden number w(k, 2, 2, ..., 2; r) is denoted by w(2)(k; r). B. Landman and A. Robertson 9] showed that for k < r < 3/2 (k - 1) and r >= 2k + 2, the inequality w(2)(k; r) <= r(k - 1) holds. In this note, we establish some results on w(2)(k; r) for 2 <= r <= k.