Fractional clique decompositions of dense graphs

For each , we show that any graph G with minimum degree at least 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, 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 number , and any , any sufficiently large graph G with minimum degree at least has, subject to some further simple necessary divisibility conditions, an (exact) F‐decomposition.