Erdős–Szekeres theorem for cyclic permutations

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 $\ell +1$.