On Monochromatic Ascending Waves

A sequence of positive integers $w_1,w_2,...,w_n$ is called an ascending wave if $w_{i+1}-w_i \geq w_i - w_{i-1}$ for $2 \leq i \leq n-1$. For integers $k,r\geq1$, 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, Erd\H{o}s, and Freedman defined such sequences and proved that $k^2-k+1\leq AW(k;2) \leq {1/3}(k^3-4k+9)$. 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 $\epsilon > 0$, $$\frac{k^{2r-1-\epsilon}}{2^{r-1}(40r)^{r^2-1}}(1+o(1)) \leq AW(k;r) \leq \frac{k^{2r-1}}{(2r-1)!}(1+o(1))$$ holds for all $r \geq 1$, which, in particular, improves upon the best known upper bound for $AW(k;2)$. Additionally, we show that for fixed $k \geq 3$, $$AW(k;r)\leq\frac{2^{k-2}}{(k-1)!} r^{k-1}(1+o(1)).$$