Approximating sparse binary matrices in the cut-norm

The cut-norm ||A||C of a real matrix A = (aij)i∈R,j∈S is the maximum, over all I ⊂ R, J ⊂ S of the quantity | ∑ i∈I,j∈J aij |. We show that there is an absolute positive constant c so that if A is the n by n identity matrix and B is a real n by n matrix satisfying ||A−B||C ≤ 1 16 ||A||C , then rank(B) ≥ cn. Extensions to denser binary matrices are considered as well.