On monochromatic ascending waves

Abstract

A sequence of positive integers w 1 , w 2 , . . . , w n is called an ascending wave if w i+1 − w i ≥ w i − w i−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 k 2 − k + 1 ≤ AW (k; 2) ≤ 1 3 (k 3 − 4k + 9). Alon and Spencer then showed that AW (k; 2) = O(k 3 ). In this article, we show that AW (k; 3) = O(k 5 ) 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 ǫ > 0,holds for all r ≥ 1, which, in particular, improves upon the best known upper bound for AW (k; 2). Additionally, we show that for fixed k ≥ 3, (1)).Soon after, Alon and Spencer [1] showed that for sufficiently large k, AW (k; 2) > k 3 10 21 − k 2 10 20 − k 10 + 4.

New Bounds on van der Waerden-type Numbers for Generalized 3-term Arithmetic Progressions

“…Ramsey properties for a variety of such families, along with a summary of what is known about w(n; r), may be found in [6]. Other recent results of this type may be found in [1], [4], [7], and [8].…”

“…The lemma is applicable at each step since hypothesis (i) holds by (8), and hypothesis (ii) holds because the largest value x to which the lemma will be applied is u By assumption, hypothesis (iii) of Lemma 1(b) holds. Hypothesis (i) holds by (8), and hypothesis (ii) holds by (9).…”