  We study the diametric problem (i.e., optimal anticodes) in the space of
permutations under the Ulam distance. That is, let $S_n$ denote the set of
permutations on $n$ symbols, and for each $\sigma, \tau \in S_n$, define their
Ulam distance as the number of distinct symbols that must be deleted from each
until they are equal. We obtain a near-optimal upper bound on the size of the
intersection of two balls in this space, and as a corollary, we prove that a
set of diameter at most $k$ has size at most $2^{k + C k^{2/3}} n! / (n-k)!$,
compared to the best known construction of size $n!/(n-k)!$. We also prove that
sets of diameter $1$ have at most $n$ elements.
