Decompositions of graphs and hypergraphs

This thesis contains various new results in the areas of design theory and edge decompositions of graphs and hypergraphs. Most notably, we give a new proof of the existence conjecture, dating back to the 19th century. For \(r\)-graphs \(F\) and \(G\), an \(F\)-decomposition of G is a collection of edge-disjoint copies of F in G covering all edges of \(G\). In a recent breakthrough, Keevash proved that every sufficiently large quasirandom \(r\)-graph G has a \(K\)\(_f\)\(^{(r)}\) -decomposition (subject to necessary divisibility conditions), thus proving the existence conjecture. We strengthen Keevash's result in two major directions: Firstly, our main result applies to decompositions into any \(r\)-graph \(F\), which generalises a fundamental theorem of Wilson to hypergraphs. Secondly, our proof framework applies beyond quasirandomness, enabling us e.g. to deduce a minimum degree version. For graphs, we investigate the minimum degree setting further. In particular, we determine the decomposition threshold' of every bipartite graph, and show that the threshold of cliques is equal to its fractional analogue. We also present theorems concerning optimal path and cycle decompositions of quasirandom graphs. This thesis is based on joint work with Daniela Kuhn and Deryk Osthus, Allan Lo and Richard Montgomery.