# 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} }

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.)

## 34 Citations

### A randomized construction of high girth regular graphs

- MathematicsRandom 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.

### On a Conjecture of Erdős on Locally Sparse Steiner Triple Systems

- MathematicsCombinatorica
- 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…

### Approximate Steiner (r − 1, r, n)‐systems without three blocks on r + 2 points

- MathematicsJournal of Combinatorial Designs
- 2019

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.…

### On a Conjecture of Erdős on Locally Sparse Steiner Triple Systems

- MathematicsComb.
- 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.

### Counting extensions revisited

- MathematicsRandom 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…

### On partial Steiner (n, r, `)-system process

- Mathematics
- 2020

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,…

### Triple systems with no three triples spanning at most five points

- MathematicsBulletin 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…

### Prague dimension of random graphs

- Mathematics, Computer ScienceArXiv
- 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.

### Extremal aspects of graph and hypergraph decomposition problems

- MathematicsBCC
- 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…

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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.

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A famous theorem of Kirkman says that there exists a Steiner triple system of order $n$ if and only if $n\equiv 1,3\mod{6}$. In 1976, Erd\H{o}s conjectured that one can find so-called `sparse'…

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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…