Yeah, I got tripped up by that formulation as well and it's actually something that annoys me with a lot of algorithms that have some properties proven in a limit: It's "easy" (or at least possible) to mathematically prove that in the limit of some variable, the property will hold: If you repeat the challenge increasingly often, the probability of being lied to will get arbitrarily close to zero; for sufficiently large input sizes, some algorithm runs in linear time; with sufficiently large amounts of training data and iterations, some prediction error will become arbitrarily small, etc etc.
But none of that is telling you how much is "sufficient", or even which order of magnitude we're talking about. If the quantity has a real life cost, this would result in enormous practical differences.
(With the formula you have given for the ZK proof, we're at least one step further: You can start with the desired probability, e.g. the gamma ray burst und calculate the required minimum k from that - also, it's easy to see that the color problem lends itself well to such proofs because the probability of failure drops exponentially quickly with growing k, so the actual k you choose can be relatively small. But if all you have is a proof in the limit, that's not possible)
But none of that is telling you how much is "sufficient", or even which order of magnitude we're talking about. If the quantity has a real life cost, this would result in enormous practical differences.
(With the formula you have given for the ZK proof, we're at least one step further: You can start with the desired probability, e.g. the gamma ray burst und calculate the required minimum k from that - also, it's easy to see that the color problem lends itself well to such proofs because the probability of failure drops exponentially quickly with growing k, so the actual k you choose can be relatively small. But if all you have is a proof in the limit, that's not possible)