Zarankiewicz's Conjecture is finite for each fixed m

Zarankiewicz's 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)/2n/2(n − 1)/2, for all positive integers m, n. This conjecture has only been verified for min{m, We determine, for each positive integer m, an integer N 0 = N 0 (m) with the following property: if cr(Km,n) = Z(m, n) for all n ≤ N 0 , then cr(Km,n) = Z(m, n) for every n. This yields, for each fixed integer m, a finite algorithm that either proves that cr(Km,n) = Z(m, n) for every n, or else finds a counterexample.