We show that given a positive integer m, a real number p ∈ [2, ∞) and 1 ≤ s < p * the set of non-multiple (r; s)-summing m-linear forms on p × · · · × p contains, except for the null vector, a closed subspace of maximal dimension whenever r < 2ms s+2m−ms . This result is optimal since for r ≥ 2ms s+2m−ms all m-linear forms on p × · · · × p are multiple (r; s)-summing. In particular, among other results, we generalize a result related to cotype (from 2010) due to Botelho et al.The classical Bohnenblust-Hille inequality [7], in the modern terminology, can be stated in terms of multiple summing operators, as remarked in [23] (see also [10]):