Erd\H{o}s-Szekeres theorem for cyclic permutations

TL;DR

This paper extends the Erd ext{H}os-Szekeres theorem to cyclic permutations, establishing tight bounds and characterizing extremal cases for increasing or decreasing cyclic sub-permutations.

Contribution

It introduces a cyclic permutation analogue of the Erd ext{H}os-Szekeres theorem, providing tight bounds and characterizations for maximum-length permutations without certain cyclic sub-permutations.

Findings

Every cyclic permutation of length (k-1)(7)(7)+2 has specific cyclic sub-permutations.

The bounds for the existence of increasing or decreasing cyclic sub-permutations are tight.

Characterization of maximum-length permutations lacking certain cyclic sub-permutations.

Abstract

We provide a cyclic permutation analogue of the Erd\H os-Szekeres theorem. In particular, we show that every cyclic permutation of length $(k-1)(\ell-1)+2$ has either an increasing cyclic sub-permutation of length $k+1$ or a decreasing cyclic sub-permutation of length $\ell+1$, and show that the result is tight. We also characterize all maximum-length cyclic permutations that do not have an increasing cyclic sub-permutation of length $k+1$ or a decreasing cyclic sub-permutation of length $\ell+1$.