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arXiv · 2308.15695

On Integer Sets Excluding Permutation Pattern Waves

Kevin Cong
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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.

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Kevin Cong. 2023-08-30. On Integer Sets Excluding Permutation Pattern Waves. https://arxiv.org/abs/2308.15695

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