(0, 1)-Matrices with No Half-Half Submatrix of Ones

We consider the minimum number of zeroes in a 2m 2n (0; 1)-matrix M that contains no m n submatrix of ones. We show that this number, denoted by f(m; n), is at least 2n + m + 1 for m n. We determine exactly when this bound is sharp and determine the extremal matrices in these cases. For any m, the bound is sharp for n = m and for all but nitely many n > m. A general upper bound due to Gentry, f(m; n) 2m + 2n ? gcd(m; n) + 1, is also derived. Our problem is a special case of the well-known Zarankiewicz problem.