  A crucial permutation is a permutation that avoids a given set of
prohibitions, but any of its extensions, in an allowable way, results in a
prohibition being introduced.
  In this paper, we introduce five natural types of crucial permutations with
respect to monotone patterns, notably quadrocrucial permutations that are
linked most closely to Erd\H{o}s-Szekeres extremal permutations. The way we
define right-crucial and bicrucial permutations is consistent with the
definition of respective permutations studied in the literature in the contexts
of other prohibitions. For each of the five types, we provide its
characterization in terms of Young tableaux via the RSK correspondence.
Moreover, we use the characterizations to prove that the number of such
permutations of length $n$ is growing when $n\to\infty$, and to enumerate
minimal crucial permutations in all but one case. We also provide other
enumerative results.
