  We study Ramsey-type problems on sets avoiding sequences whose consecutive
differences have a fixed relative order. For a given permutation $\pi \in S_k$,
a $\pi$-wave is a sequence $x_1 < \cdots < x_{k+1}$ such that $x_{i+1} - x_i >
x_{j+1} - x_j$ if and only if $\pi(i) > \pi(j)$. A subset of $[n] =
\{1,\ldots,n\}$ is $\pi$-wave-free if it does not contain any $\pi$-wave. Our
first main result shows that the size of the largest $\pi$-wave-free subset of
$[n]$ is $O\left((\log n)^{k-1}\right)$. We then classify all permutations for
which this bound is tight. In the cases where it is not tight, we prove
stronger polylogarithmic upper bounds. We then apply these bounds to a closely
related coloring problem studied by Landman and Robertson.
