#### Vol. 12, No. 2, 2019

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Erdős–Szekeres theorem for cyclic permutations

### Éva Czabarka and Zhiyu Wang

Vol. 12 (2019), No. 2, 351–360
DOI: 10.2140/involve.2019.12.351
##### Abstract

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

##### Keywords
cyclic Erdős–Szekeres theorem
Primary: 05D99