Rational decomposition of dense hypergraphs and some related eigenvalue estimates

We consider the problem of decomposing some family of t-subsets, or t-uniform hypergraph G, into copies of another, say H, with nonnegative rational weights. For fixed H on k vertices, we show that this is always possible for all G having sufficiently many vertices and density at least 1-C(t)k-2t. In particular, for the case t=2, all large graphs with density at least 1-2k-4 admit a rational decomposition into cliques Kk. The proof relies on estimates of certain eigenvalues in the Johnson scheme. The concluding section discusses some applications to design theory and statistics, as well as some relevant open problems.