Fractional clique decompositions of dense graphs

For each r ≥ 4 , we show that any graph G with minimum degree at least ( 1 − 1 / ( 100 r ) ) | 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, 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 χ ( F ) ≥ 4 , and any ε > 0 , any sufficiently large graph G with minimum degree at least ( 1 − 1 / ( 100 χ ( F ) ) + ε ) | G | has, subject to some further simple necessary divisibility conditions, an (exact) F ‐decomposition.