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A famous theorem of Kirkman says that there exists a Steiner triple system of order n if and only if n≡1,3mod6. 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.
ASJC Scopus subject areas
- Discrete Mathematics and Combinatorics
- Computational Mathematics
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- 1 Finished
1/11/16 → 31/10/18
Project: Research Councils