An Erd\H{o}s--Hajnal analogue for permutation classes

TL;DR

This paper proves that permutation classes excluding all layered or colayered permutations necessarily contain long monotone subsequences proportional to their length.

Contribution

It establishes an Erdős–Hajnal type result for permutation classes with specific exclusion conditions.

Findings

Existence of a constant c for long monotone subsequences

Permutation classes avoiding layered and colayered permutations have linear monotone subsequences

Provides a new structural insight into permutation classes

Abstract

Let $\mathcal{C}$ be a permutation class that does not contain all layered permutations or all colayered permutations. We prove that there is a constant $c$ such that every permutation in $\mathcal{C}$ of length $n$ contains a monotone subsequence of length $cn$.