  For any group G of order n, a subset A of G is said to be product-free if
there is no solution of the equation ab=c with a,b,c in A. Previous results of
Gowers showed that the size of any product-free subset of G is at most
n/d^(1/3), where d is the smallest dimension of a nontrivial representation of
G. However, this upper bound does not match the best lower bound. We will
generalize the upper bound to the case of product-poor subsets A, in which the
equation ab=c is allowed to have a few solutions with a,b,c in A. We prove that
the upper bound for the size of product-poor subsets matches the best lower
bound in many families of groups. We will also generalize the concept of
product-free to the case in which we have many subsets of a group, and
different constraints about products of the elements in the subsets.
