From a followup post "Progress checking Zarankiewicz’s conjecture on the brick factory problem" [http://mappingignorance.org/2013/10/01/progress-checking-zar...]

"This means that N_0(5) \approx 6\cdot 10^{1228} , a really big number. The fastest computers today perform in the order of petaFLOPS (10^{15} operations per second), so even if they could check each case in a single operation, it would take in the order of 10^{1204} years to check if the conjecture holds for K_{5,n} with all n\leq N_0 ."

I guess the tangle of cables behind my desk isn't going to improve anytime soon, then.