Partitioning permutations into monotone subsequences

TL;DR

This paper disproves Barber's conjecture by constructing permutations where all small subsequences are $k$-coverable but the entire permutation is not, for all $k \, \ge \, 3$.

Contribution

It provides counterexamples to Barber's conjecture, showing the conjecture does not hold for any $k \ge 3$.

Findings

Counterexamples exist for all $k \ge 3$.

Small subsequences are $k$-coverable while the whole permutation is not.

Disproves the conjecture that local $k$-coverability implies global $k$-coverability.

Abstract

A permutation is $k$-coverable if it can be partitioned into $k$ monotone subsequences. Barber conjectured that, for any given permutation, if every subsequence of length $k+2 \choose 2$ is $k$-coverable then the permutation itself is $k$-coverable. This conjecture, if true, would be best possible. Our aim in this paper is to disprove this conjecture for all $k \ge 3$. In fact, we show that for any $k$ there are permutations such that every subsequence of length at most $(k/6)^{2.46}$ is $k$-coverable while the permutation itself is not.