(0, 1)-matrices with no half-half submatrix of ones (Q1372611)

If \(m\), \(n\) are arbitrary positive integers, let \(f(m,n)\) denote the minimum number of zeroes in a \(2m\times 2n\) matrix whose all entires are \(0\) or \(1\), and which contains no \(m\times n\) submatrix of ones. The authors prove that then \(f(m,n)\leq 2m+ 2n-\text{gcd}(m, n)+1\) and \(f(m, n)\geq m+2n+ 1\) for \(m\leq n\). If \(m\leq n\) and \(m\) is fixed, they also prove that \(f(m,n)= m+2n+1\) if and only if \(n= m\) or all but finitely many \(n\) are greater than \(m\).