  A subset P of N x N is called Schur bounded if every infinite matrix with
bounded entries which is zero off of P yields a bounded Schur multiplier on
B(H). Such sets are characterized as being the union of a subset with at most k
entries in each row with another that has at most k entries in each column, for
some finite k. If k is optimal, there is a Schur multiplier supported on the
pattern with norm O(k^(1/2)), which is sharp up to a constant. The same
techniques give a new, more elementary proof of results of Varopoulos and
Pisier on Schur multipliers with given matrix entries of random sign. We
consider the Schur multipliers for certain matrices which have a large symmetry
group. In these examples, we are able to compute the Schur multiplier norm
exactly. This is carried out in detail for a few examples including the Kneser
graphs.
