Asymptotically good coverings

The Erdos-Hanani conjecture is that for fixed r < k and n large there exists a covering of all r-sets of an n-set by a family of k-sets whose cardinality is asymptotic (in n) to the "counting" lower bound. This conjecture was first proven by Rodl, here we give a more direct argument. We use probabilistic methods, selecting k-sets in large groups, and showing that the hypergraph of uncovered r-sets retains a property we call quasirandomness, meaning that it has the essential (for us) properties of random hypergraph.