Fractional clique decompositions of dense graphs

For each r >= 4, we show that any graph G with minimum degree at least (1-1/(100r))|G| has a fractional K-r-decomposition. This improves the best previous bounds on the minimum degree required to guarantee a fractional K-r-decomposition given by Dukes (for small r) and Barber, Kuhn, 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 Turan graphs). In combination with work by Glock, Kuhn, Lo, Montgomery, and Osthus, this shows that, for any graph F with chromatic number chi(F)>= 4, and any epsilon>0, any sufficiently large graph G with minimum degree at least (1-1/(100 chi(F))+epsilon)|G| has, subject to some further simple necessary divisibility conditions, an (exact) F-decomposition.