Zarankiewiczʼs Conjecture is finite for each fixed m

Zarankiewiczʼs Crossing Number Conjecture states that the crossing number cr(Km,n) of the complete bipartite graph Km,n equals Z(m,n):=⌊m/2⌋⌊(m−1)/2⌋⌊n/2⌋⌊(n−1)/2⌋, for all positive integers m, n. This conjecture has only been verified for min{m,n}⩽6, for K7,7, K7,8, K7,9, and K7,10, and for K8,8, K8,9, and K8,10. We determine, for each positive integer m, an integer N0=N0(m) with the following property: if cr(Km,n)=Z(m,n) for all n⩽N0, then cr(Km,n)=Z(m,n) for every n. This yields, for each fixed integer m, a finite algorithm that either shows that the assertion: “for all n, cr(Km,n)=Z(m,n)” is true, or else finds a counterexample.