The iterative absorption method has recently led to major progress in the area of (hyper-)graph decompositions. Amongst other results, a new proof of the Existence conjecture for combinatorial designs, and some generalizations, was obtained. Here, we illustrate the method by investigating triangle decompositions: we give a simple proof that a triangle-divisible graph of large minimum degree has a triangle decomposition and prove a similar result for quasirandom host graphs.

A famous theorem of Kirkman says that there exists a Steiner triple system of order n if and only if n ≡ 1, 3 mod 6. In 1973, Erdős conjectured that one can find so-called 'sparse' Steiner triple systems. Roughly speaking, the aim is to have at most j − 3 triples on every set of j points, which would be best possible. (Triple systems with this sparseness property are also referred to as having high girth.) We prove this conjecture asymptotically by analysing a natural generalization of the triangle removal process. Our result also solves a problem posed by Lefmann, Phelps and Rödl as well as Ellis and Linial in a strong form, and answers a question of Krivelevich, Kwan, Loh, and Sudakov. Moreover, we pose a conjecture which would generalize the Erdős conjecture to Steiner systems with arbitrary parameters and provide some evidence for this.

We study the F -decomposition threshold δF for a given graph F . Here an F

No abstract

Where a licence is displayed above, please note the terms and conditions of the licence govern your use of this document.When citing, please reference the published version. Take down policyWhile the University of Birmingham exercises care and attention in making items available there are rare occasions when an item has been uploaded in error or has been deemed to be commercially or otherwise sensitive.