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.
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We are grateful to Tom Bohman and Lutz Warnke for pointing out a minor oversight in the calculation of Egain in an earlier version of this paper.
The research leading to these results was partially supported by the EPSRC, grant nos. EP/N019504/1 (D. Kühn) and EP/P002420/1 (A. Lo), by the Royal Society and the Wolfson Foundation (D. Kühn) as well as by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013) / ERC Grant Agreement no. 306349 (S. Glock and D. Osthus).
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Glock, S., Kühn, D., Lo, A. et al. On a Conjecture of Erdős on Locally Sparse Steiner Triple Systems. Combinatorica 40, 363–403 (2020). https://doi.org/10.1007/s00493-019-4084-2
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