Our main result is that every graph G on n≥ 10 4 r 3 vertices with minimum degree δ
(G)≥(1− 1/10 4 r 3/2) n has a fractional K r-decomposition. Combining this result with recent
work of Barber, Kühn, 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 ck> 0 such that every k-uniform hypergraph G on n vertices with …

For each, we show that any graph G with minimum degree at least has a fractional Kr‐
decomposition. This improves the best previous bounds on the minimum degree required to
guarantee a fractional Kr‐decomposition given by Dukes (for small r) and Barber, Kühn, Lo,
Montgomery, and Osthus (for large r), giving the first bound that is tight up to the constant
multiple of r (seen, for example, by considering Turán graphs). In combination with work by
Glock, Kühn, Lo, Montgomery, and Osthus, this shows that, for any graph F with chromatic …