On the half?half case of the Zarankiewicz problem

Consider the minimum number f(m,n) of zeroes in a 2m×2n (0, 1)-matrixM that contains no m×n submatrix of ones. This special case of the well-known Zarankiewicz problem was studied by Griggs and Ouyang, who showed, for m ≤ n, that 2n+m+1 ≤ f(m,n) ≤ 2n + 2m − gcd(m,n) + 1. The lower bound is sharp when m is fixed for all large n. They proposed determining limm→∞{f(m,m+ 1)/m}. In this paper, we show that this limit is 3. Indeed, we determine the actual value of f(m, km + 1) for all k, m. For general m, n, we derive a new upper bound on f(m,n). We also give the actual value of f(m,n) for all m ≤ 7 and n ≤ 20. Running head: The Half-Half Problem 1 Research supported in part by grant NSA/MSP MDA904–95H1024. 2 Research supported in part by grant NSF DMS–9701211. The Half-Half Problem Section