# Large girth approximate Steiner triple systems

@article{Bohman2018LargeGA,
title={Large girth approximate Steiner triple systems},
author={Tom Bohman and Lutz Warnke},
journal={Journal of the London Mathematical Society},
year={2018},
volume={100}
}
• Published 3 August 2018
• Mathematics
• Journal of the London Mathematical Society
In 1973, Erdős asked whether there are n ‐vertex partial Steiner triple systems with arbitrary high girth and quadratically many triples. (Here girth is defined as the smallest integer g⩾4 for which some g ‐element vertex‐set contains at least g−2 triples.)
• Mathematics
Random Struct. Algorithms
• 2021
It is shown that with high probability this algorithm yields a $k-regular graph with girth at least$g$and implies that there are$\left( \Omega (n) \right)^{kn/2}$labeled$k$-regular$n$-vertex graphs with g diameter at least g. • Mathematics Combinatorica • 2020 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 For a family F of r ‐graphs, let ex(n,F) denote the maximum number of edges in an F ‐free r ‐graph on n vertices. Let Fr(v,e) denote the family of all r ‐graphs with e edges and at most v vertices. • Mathematics Comb. • 2020 This work proves the Erdős conjecture that one can find so-called ‘sparse’ Steiner triple systems with arbitrary parameters asymptotically by analysing a natural generalization of the triangle removal process. • Mathematics Random Struct. Algorithms • 2022 We consider rooted subgraphs in random graphs, that is, extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given For given integers r and  such that 2 ≤  ≤ r − 1, an r-uniform hypergraph H is called a partial Steiner (n, r, )-system, if every subset of size  lies in at most one edge of H. In particular, • Stefan Glock • Mathematics Bulletin of the London Mathematical Society • 2018 We show that the maximum number of triples on n points, if no three triples span at most five points, is (1±o(1))n2/5 . More generally, let f(r)(n;k,s) be the maximum number of edges in an r ‐uniform • Mathematics, Computer Science ArXiv • 2020 It is shown that the Prague dimension of the binomial random graph is typically of order n/log n for constant edge-probabilities, i.e., edges of size O(log n) for random hypergraphs with large uniformities. • Mathematics BCC • 2021 We survey recent advances in the theory of graph and hypergraph decompositions, with a focus on extremal results involving minimum degree conditions. We also collect a number of intriguing open ## References SHOWING 1-10 OF 37 REFERENCES • A. Wolfe • Mathematics Electron. J. Comb. • 2005 This paper gives a proof that the spectrum of orders of 5-sparse Steiner triple systems has arithmetic density$1$as compared to the admissible orders. • Mathematics, Computer Science J. Comb. Theory, Ser. A • 2007 • Mathematics • 2000 We show that an anti-Pasch Steiner triple system of order$v$exists for$v\equiv 1$or 3 (mod 6), apart from$v=7$and 13. We give estimates on the number of combinatorial designs, which prove (and generalise) a conjecture of Wilson from 1974 on the number of Steiner Triple Systems. This paper also serves as an • Mathematics Comb. • 2020 This work proves the Erdős conjecture that one can find so-called ‘sparse’ Steiner triple systems with arbitrary parameters asymptotically by analysing a natural generalization of the triangle removal process. • Mathematics, Computer Science ArXiv • 2010 It is shown that with high probability the number of edges in the final graph is at most O(n/4) -log^{5/4}n, while the random greedy algorithm for constructing a large partial Steiner-Triple-System is defined as follows. • Mathematics • 2018 A famous theorem of Kirkman says that there exists a Steiner triple system of order$n$if and only if$n\equiv 1,3\mod{6}$. In 1976, Erd\H{o}s conjectured that one can find so-called sparse' Three constructions for anti‐mitre Steiner triple systems and a construction for 5‐sparse ones are presented and it is shown that one can construct anti-mitre systems for at least 13/14 of the admissible orders. • Mathematics Electron. J. Comb. • 2014 A random$r-uniform Cayley' hypergraph on $S_n$ which has girth $\Omega (n^{1/3})$ with high probability, in contrast to random regular $r$- uniform hypergraphs, which have constant girth with positive probability.
• Mathematics
• 1993
In this article Turan-type problems for several triple systems arising from (k, k − 2)-configurations [i.e. (k − 2) triples on k vertices] are considered. It will be shown that every Steiner triple