Let $f(k)$ be the minimal $n$ such that any $2$-colouring of $\{1,\dots ,n\}$ contains a monochromatic $k$-term descending wave: a sequence $x_1<\cdots <x_k$ such that, for $1<j<k$,$$x_j\ge \frac{x_{j+1}+x_{j-1}}{2}.$$Estimate $f(k)$. In particular is it true that $f(k)=k^2-k+1$ for all $k$?