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 $cn$.
Splittability and 1-amalgamability of permutation classes
On the growth of permutation classes
Enumeration of Permutation Classes and Weighted Labelled Independent Sets
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