  We view the RSK correspondence as associating to each permutation $\pi \in
S_n$ a Young diagram $\lambda=\lambda(\pi)$, i.e. a partition of $n$. Suppose
now that $\pi$ is left-multiplied by $t$ transpositions, what is the largest
number of cells in $\lambda$ that can change as a result? It is natural refer
to this question as the search for the Lipschitz constant of the RSK
correspondence.
  We show upper bounds on this Lipschitz constant as a function of $t$. For
$t=1$, we give a construction of permutations that achieve this bound exactly.
For larger $t$ we construct permutations which come close to matching the upper
bound that we prove.
