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

We consider the minimum number of zeroes in a 2m× 2n(0,1)-matrixMthat contains nom×nsubmatrix of ones. We show that this number, denoted byf(m,n), is at least 2n+m+ 1 form≤n. We determine exactly when this bound is sharp and determine the extremal matrices in these cases. For anym, the bound is sharp forn=mand for all but finitely manyn>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.