Fractional clique decompositions of dense graphs

For each $r\ge 4$, we show that any graph $G$ with minimum degree at least $(1-1\mathrm{/}100r)\mathrm{\mid }G\mathrm{\mid }$ 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, K"uhn, 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 Tur'an graphs). In combination with work by Glock, K"uhn, Lo, Montgomery and Osthus, this shows that, for any graph $F$ with chromatic number $\chi (F)\ge 4$, and any $\epsilon >0$, any sufficiently large graph $G$ with minimum degree at least $(1-1\mathrm{/}100\chi (F)+\epsilon )\mathrm{\mid }G\mathrm{\mid }$ has, subject to some further simple necessary divisibility conditions, an (exact) $F$-decomposition.