  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.
