Rational decomposition of dense hypergraphs and some related eigenvalue estimates

Details Rational decomposition of dense hypergraphs and some related eigenvalue estimates Rational decomposition of dense hypergraphs and some related eigenvalue estimates

2011-08-07

arxiv.org/abs/1108.1576v1

Mathematics

Combinatorics

17 pages

Scientific paper

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 $K_k$. 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.

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