Fractional Clique Decompositions of Dense Graphs and Hypergraphs

Our main result is that every graph \(G\) on \(n\ge 10^4r^3\) vertices with minimum degree \(\delta(G) \ge (1 - 1 / 10^4 r^{3/2} ) n\) has a fractional \(K_r\)-decomposition. Combining this result with recent work of Barber, K\"uhn, 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 \(c_k>0\) such that every \(k\)-uniform hypergraph \(G\) on \(n\) vertices with minimum codegree at least \((1- c_k /r^{2k-1}) n \) has a fractional \(K^{(k)}_r\)-decomposition, where \(K^{(k)}_r\) is the complete \(k\)-uniform hypergraph on \(r\) vertices. (Related fractional decomposition results for triangles have been obtained by Dross and for hypergraph cliques by Dukes as well as Yuster.) All the above new results involve purely combinatorial arguments. In particular, this yields a combinatorial proof of Wilson's theorem that every large \(F\)-divisible complete graph has an \(F\)-decomposition.