# A sharp threshold for random graphs with a monochromatic triangle in every edge coloring

@inproceedings{Friedgut2006AST, title={A sharp threshold for random graphs with a monochromatic triangle in every edge coloring}, author={Ehud Friedgut and Vojtech R{\"o}dl and Andrzej Rucinski and Prasad Tetali}, booktitle={Memoirs of the American Mathematical Society}, year={2006} }

Let R be the set of all finite graphs G with the Ramsey property that every coloring of the edges of G by two colors yields a monochromatic triangle. In this paper we establish a sharp threshold for random graphs with this property. Let G(n,p) be the random graph on n vertices with edge probability p. We prove that there exists a function b =b(n) = �(1) such that for any " > 0, as n tends to infinity,

## 37 Citations

Sharp thresholds for Ramsey properties of strictly balanced nearly bipartite graphs

- MathematicsRandom Struct. Algorithms
- 2018

A simpler proof of Friedgut's criteria (1999) for sharp thresholds, and on the recently developed container method for independent sets in hypergraphs by Saxton and Thomason, and Balogh, Morris and Samotij are obtained.

Upper tails for subgraph counts in random graphs

- Mathematics
- 2004

LetG be a fixed graph and letXG be the number of copies ofG contained in the random graphG(n, p). We prove exponential bounds on the upper tail ofXG which are best possible up to a logarithmic factor…

Ramsey properties of random graphs and hypergraphs

- Mathematics
- 2013

The topic of this thesis are Ramsey-type problems in random graphs and hypergraphs. Ramsey theory has its origins in a famous 1930 paper by Frank P. Ramsey. Loosely speaking its object of study are…

Ramsey Properties of Random k-Partite, k-Uniform Hypergraphs

- MathematicsSIAM J. Discret. Math.
- 2007

The threshold probability for the property that every coloring of the edges of a random binomial of a uniform hypergraph yields a monochromatic copy of some fixed hypergraph is investigated.

Two variants of the size Ramsey number

- MathematicsDiscuss. Math. Graph Theory
- 2005

The first part of this paper shows that when H is a complete graph Kk on k vertices, then minf (H, r) = (R − 1)/2, where R = R(k; r) is the classical Ramsey number.

The Online Clique Avoidance Game on Random Graphs

- MathematicsAPPROX-RANDOM
- 2005

It is shown that there is a function N0 = N0(l, n) such that the player can asymptotically almost surely survive up to N(n) ≪ N0 edges by playing greedily and that this is best possible, i.e., there is no strategy such thatThe player's object is to color as many edges as possible without creating a monochromatic clique of some fixed size l.

A randomized version of Ramsey's theorem

- MathematicsRandom Struct. Algorithms
- 2011

This work determines the threshold for a random k-uniform hypergraph with n vertices to be r-Ramsey-forcing and settles an open question from Allen, Bottcher, Hladký, and Piguet.

Ramsey properties of random discrete structures

- MathematicsRandom Struct. Algorithms
- 2010

The threshold for Rado's theorem for solutions of partition regular systems of equations in random subsets of the integers is determined and the 1-statement of the conjectured threshold for Ramsey’s theorem for random hypergraphs is proved.

Ramsey and Universality Properties of Random Graphs

- Mathematics
- 2016

This thesis investigates the interplay between two branches of discrete mathematics: Ramsey theory and random graphs. The origins of Ramsey theory can already be found in the work of Hilbert from the…

## References

SHOWING 1-10 OF 63 REFERENCES

Random Graphs with Monochromatic Triangles in Every Edge Coloring

- MathematicsRandom Struct. Algorithms
- 1994

We prove that for every r ⩾ 2 there is C > 0 such that if pCn−1/2 then almost surely every r-coloring of the edges of the binomial random graph K(n, p) results in a monochromatic triangle. © 1994…

Sharp Thresholds for Ramsey Properties of Random Graphs

- Mathematics
- 1999

In a series of papers culminating in 11] RR odl, Ruci nski and others study the thresholds of Ramsey properties of random graphs and hy-pergraphs, i.e. for a given graph H, when does a random graph…

Large triangle-free subgraphs in graphs withoutK4

- MathematicsGraphs Comb.
- 1986

It is shown that for arbitrary positiveε there exists a graph withoutK4 and so that all its subgraphs containing more than 1/2 +ε portion of its edges contain a triangle (Theorem 2), and it is proved that such graphs have necessarily low edge density.

Ramsey Properties of Random Hypergraphs

- MathematicsJ. Comb. Theory, Ser. A
- 1998

A recent version of the hypergraph regularity lemma due to Frankl and Rodl is utilized, finding the threshold for the property that everyr-coloring of the vertices of K(k)(n,p) results in a monochromatic copy of G.

Upper tails for subgraph counts in random graphs

- Mathematics
- 2004

LetG be a fixed graph and letXG be the number of copies ofG contained in the random graphG(n, p). We prove exponential bounds on the upper tail ofXG which are best possible up to a logarithmic factor…

Sharp thresholds of graph properties, and the -sat problem

- Mathematics
- 1999

Consider G(n, p) to be the probability space of random graphs on n vertices with edge probability p. We will be considering subsets of this space defined by monotone graph properties. A monotone…

Threshold functions for Ramsey properties

- Mathematics
- 1995

Probabilistic methods have been used to approach many problems of Ramsey theory. In this paper we study Ramsey type questions from the point of view of random structures. Let K(n, N) be the random…

Two variants of the size Ramsey number

- MathematicsDiscuss. Math. Graph Theory
- 2005

The first part of this paper shows that when H is a complete graph Kk on k vertices, then minf (H, r) = (R − 1)/2, where R = R(k; r) is the classical Ramsey number.

A Sharp Threshold for k-Colorability

- MathematicsRandom Struct. Algorithms
- 1999

It is concluded that for any given value of n the the chromatic number of G(n; d=n) is concentrated in one value for all but a small fraction of d > 1.