Numerical experiments suggest that cooperation is a very rare phenomenon in autocatalytic networks with more than about five interacting species. There is strong evidence that the probability for cooperative behavior in random networks decreases at least exponentially and at most as an2 with the number n of interacting species. The evolution of larger persistent networks by subsequent incorporation of additional species into a cooperative network is shown to be even less probable. This leads to the conclusion that hypercycle-type interactions alone are not sufficient to maintain the genetic information of more than a few interacting species in a pure RNA world.