HYPERGRAPH COVERINGS AND LOCAL COLORINGS

If the edge sets of some r-uniform hypergraphs H1, ..., Ht contain all the r-tuples of an n-element set, and each Hi has chromatic number at most k, then for the sum of the orders of the Hi we have Σ|V(Hi)| ≥ (n log ( n (r - 1))) log k. In particular, if Kn r has an edge coloring such that each vertex is incident to edges of at most s colors and every monochromatic class of edges forms a k-vertex-colorable hypergraph, then n ≤ (r - 1) ks. Both bounds are sharp. © 1991.