On Monochromatic Ascending Waves

A sequence of positive integers w1, w2, . . . , wn is called an ascending wave if wi+1 − wi ≥ wi − wi−1 for 2 ≤ i ≤ n − 1. For integers k, r ≥ 1, let AW (k; r) be the least positive integer such that under any r-coloring of [1, AW (k; r)] there exists a k-term monochromatic ascending wave. The existence of AW (k; r) is guaranteed by van der Waerden’s theorem on arithmetic progressions since an arithmetic progression is, itself, an ascending wave. Originally, Brown, Erdős, and Freedman defined such sequences and proved that k2 − k + 1 ≤ AW (k; 2) ≤ 3(k 3 − 4k + 9). Alon and Spencer then showed that AW (k; 2) = Θ(k3). In this article, we show that AW (k; 3) = Θ(k5) as well as offer a proof of the existence of AW (k; r) independent of van der Waerden’s theorem. Furthermore, we prove that for any ! > 0 and any fixed r ≥ 1, k2r−1−! 2r−1(40r)r−1 (1 + o(1)) ≤ AW (k; r) ≤ k 2r−1 (2r − 1)! + o(1)), which, in particular, improves upon the best known upper bound for AW (k; 2). Additionally, we show that for fixed k ≥ 3, AW (k; r) ≤ 2 k−2 (k − 1)! r k−1(1 + o(1)).