Hypergraph coverings and local colorings (Q1121913)

If the edge sets of some r-uniform hypergraphs \b{H}\({}_ 1,...,\underline H_ t\) contain all the r-tuples of an n-element set, and each \b{H}\({}_ i\) has chromatic number at most k, then for the sum of the orders of the \b{H}\({}_ i\) we have \(\sum | V(\underline H_ i)| \geq (n \log (n/r-1))/\log k\). In particular, if \(K^ r_ n\) 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 \(n\leq (r-1)k^ s\). Both bounds are sharp. The first inequality can be extended for more general weight functions f: \({\mathbb{R}}^+\to {\mathbb{R}}^+\), f convex, f(x)/x decreasing for \(x>0\); in this case \(\sum f(| V(\underline H_ i)|)\geq (f(n)\log (n/r- 1))/\log k.\)