  Our main result is that every graph $G$ on $n\ge 10^4r^3$ vertices with
minimum degree $\delta(G) \ge (1 - 1 / 10^4 r^{3/2} ) n$ has a fractional
$K_r$-decomposition. Combining this result with recent work of Barber, K\"uhn,
Lo and Osthus leads to the best known minimum degree thresholds for exact
(non-fractional) $F$-decompositions for a wide class of graphs~$F$ (including
large cliques). For general $k$-uniform hypergraphs, we give a short argument
which shows that there exists a constant $c_k>0$ such that every $k$-uniform
hypergraph $G$ on $n$ vertices with minimum codegree at least $(1- c_k
/r^{2k-1}) n $ has a fractional $K^{(k)}_r$-decomposition, where $K^{(k)}_r$ is
the complete $k$-uniform hypergraph on $r$ vertices. (Related fractional
decomposition results for triangles have been obtained by Dross and for
hypergraph cliques by Dukes as well as Yuster.) All the above new results
involve purely combinatorial arguments. In particular, this yields a
combinatorial proof of Wilson's theorem that every large $F$-divisible complete
graph has an $F$-decomposition.
