On the half–half case of the Zarankiewicz problem

Abstract

Consider the minimum number f(m,n) zeroes in a 2m×2n (0,1) -matrix M 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 lim m→∞{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.