More-Than-Nearly-Perfect Packings and Partial Designs

Results of Frankl and RR odl and of Pippenger and Spencer show that uniform hypergraphs which are almost regular and have small maximal pair degrees (codegrees) contain collections of pairwise disjoint edges (packings) which cover all but o(n) of the n vertices. Here we show, in particular, that regular uniform hypergraphs for which the ratio of degree to maximum codegree is n " , for some " > 0, have packings which cover all but n 1? vertices, where = (") > 0. The proof is based on the analysis of a generalized version of RR odl's nibble technique. We apply the result to the problem of nding partial Steiner systems with almost enough blocks to be Steiner systems, where we prove that, for xed positive integers t < k, there exist partial S(t; k; n)'s with at most n t?1=(2(k t)?1)+o(1) uncovered t-sets, improving the earlier o(n t) result.