# Random triangles in random graphs

@article{Heckel2018RandomTI, title={Random triangles in random graphs}, author={Annika Heckel}, journal={arXiv: Combinatorics}, year={2018} }

In a recent paper, Oliver Riordan shows that for $r \ge 4$ and $p$ up to and slightly larger than the threshold for a $K_r$-factor, the hypergraph formed by the copies of $K_r$ in $G(n,p)$ contains a copy of the binomial random hypergraph $H=H_r(n,\pi)$ with $\pi \sim p^{r \choose 2}$. For $r=3$, he gives a slightly weaker result where the density in the random hypergraph is reduced by a constant factor. Recently, Jeff Kahn announced an asymptotically sharp bound for the threshold in Shamir's… Expand

#### 7 Citations

Random cliques in random graphs

- Mathematics
- 2018

We show that for each $r\ge 4$, in a density range extending up to, and slightly beyond, the threshold for a $K_r$-factor, the copies of $K_r$ in the random graph $G(n,p)$ are randomly distributed,… Expand

Asymptotics for Shamir's Problem

- Mathematics
- 2019

For fixed $r\geq 3$ and $n$ divisible by $r$, let ${\mathcal H}={\mathcal H}^r_{n,M}$ be the random $M$-edge $r$-graph on $V=\{1,\ldots ,n\}$; that is, ${\mathcal H}$ is chosen uniformly from the… Expand

Hitting times for Shamir’s problem

- Mathematics
- Transactions of the American Mathematical Society
- 2021

For fixed $r\geq 3$ and $n$ divisible by $r$, let ${\mathcal H}={\mathcal H}^r_{n,M}$ be the random $M$-edge $r$-graph on $V=\{1,\ldots ,n\}$; that is, ${\mathcal H}$ is chosen uniformly from the… Expand

Thresholds versus fractional expectation-thresholds

- Mathematics, Computer Science
- Annals of Mathematics
- 2021

The 'axial' version of the random multi-dimensional assignment problem (earlier considered by Martin--M\'{e}zard--Rivoire and Frieze--Sorkin) is resolved and the Erd\H{o}s--Rado 'Sunflower Conjecture' is solved. Expand

Spanning $F$-cycles in random graphs

- Mathematics
- 2021

We extend a recent argument of Kahn, Narayanan and Park (Proceedings of the AMS, to appear) about the threshold for the appearance of the square of a Hamilton cycle to other spanning structures. In… Expand

A note on spanning Kr-cycles in random graphs

- Mathematics
- 2019

We find a close approximation to the threshold for the existence of a collection of edge disjoint copies of $K_r$ that form a cyclic structure and span all vertices of $G_{n,p}$. We use a recent… Expand

Hamilton Cycles in Random Graphs: a bibliography

- Mathematics
- 2019

We provide an annotated bibliography for the study of Hamilton cycles in random graphs and hypergraphs.

#### References

SHOWING 1-5 OF 5 REFERENCES

Random cliques in random graphs

- Mathematics
- 2018

We show that for each $r\ge 4$, in a density range extending up to, and slightly beyond, the threshold for a $K_r$-factor, the copies of $K_r$ in the random graph $G(n,p)$ are randomly distributed,… Expand

Factors in random graphs

- Mathematics
- 2008

Let H be a fixed graph on v vertices. For an n-vertex graph G with n divisible by v, an H-factor of G is a collection of n-v copies of H whose vertex sets partition V (G).
In this work, we… Expand

Two moments su ce for Poisson approx-imations: the Chen-Stein method

- Mathematics
- 1989

Convergence to the Poisson distribution, for the number of occurrences of dependent events, can often be established by computing only first and second moments, but not higher ones. This remarkable… Expand

Fundamentals of Stein's method

- Mathematics
- 2011

This survey article discusses the main concepts and techniques of Stein's method for distributional approximation by the normal, Poisson, exponential, and geometric distributions, and also its… Expand