On the chromatic number of set systems

An r l -system is an r-uniform hypergraph in which every set of l vertices lies in at most one edge. Let mk r l be the minimum number of edges in an r l -system that is not k-colorable. Using probabilistic techniques, we prove that ar l kr−1 ln k l/ l−1 ≤ mk r l ≤ br l kr−1 ln k l/ l−1 where br l is explicitly defined and ar l is sufficiently small. We also give a different argument proving (for even k) mk r l ≥ ar lk r−1 l/ l−1 where ar l = r − l + 1 /r 2r−1re −l/ l−1 . Our results complement earlier results of Erdős and Lovász [10] who mainly focused on the case l = 2 k fixed, and r large. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 19, 87–98, 2001