On Integer Sets Excluding Permutation Pattern Waves

TL;DR

This paper investigates the maximum size of subsets of integers avoiding specific permutation-based difference patterns, establishing bounds and classifications, and applying results to related coloring problems.

Contribution

It introduces bounds on the size of permutation wave-free sets and classifies permutations where these bounds are tight, extending understanding of pattern-avoiding sets.

Findings

Largest wave-free subset size is O((log n)^{k-1})

Classification of permutations with tight bounds

Stronger bounds for non-tight cases

Abstract

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.