A conjecture of Erdős on graph Ramsey numbers

@article{Sudakov2011ACO,
  title={A conjecture of Erdős on graph Ramsey numbers},
  author={Benny Sudakov},
  journal={Advances in Mathematics},
  year={2011},
  volume={227},
  pages={601-609}
}
  • B. Sudakov
  • Published 30 January 2010
  • Mathematics
  • Advances in Mathematics

Around a conjecture of ErdH{o}s on graph Ramsey numbers

For given graphs G1 and G2 the Ramsey number R(G1,G2), is the smallest positive integer n such that each blue-red edge coloring of the complete graph Kn contains a blue copy of G1 or a red copy of

On two problems in graph Ramsey theory

TLDR
This work improves the upper bound on the existence of a constant c such that, for any graph H on n vertices, rind(H) ≤ 2cnlogn, and moves a step closer to proving this conjecture.

On the multicolor Ramsey number of a graph with m edges

Recent developments in graph Ramsey theory

TLDR
There has been a great deal of recent progress on the study of Ramsey numbers and their variants, spurred on by the many advances across extremal combinatorics.

Ramsey numbers of sparse digraphs

Burr and Erdős in 1975 conjectured, and Chvátal, Rödl, Szemerédi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we

Ramsey numbers with prescribed rate of growth

. Let R ( G ) be the two-colour Ramsey number of a graph G . In this note, we prove that for any non-decreasing function n 6 f ( n ) 6 R ( K n ), there exists a sequence of connected graphs ( G n ) n

Chromatic number, clique subdivisions, and the conjectures of Hajós and Erdős-Fajtlowicz

TLDR
The main ingredient in the proof, which might be of independent interest, is an estimate on the order of the largest clique subdivision which one can find in every graph on n vertices with independence number α.

Ramsey numbers of degenerate graphs

A graph is $d$-degenerate if all its subgraphs have a vertex of degree at most $d$. We prove that there exists a constant $c$ such that for all natural numbers $d$ and $r$, every $d$-degenerate graph

Erdos-Hajnal-type theorems in hypergraphs

Small Ramsey Numbers

We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are

References

SHOWING 1-10 OF 24 REFERENCES

On Ramsey Numbers of Sparse Graphs

TLDR
It is shown that, for every , sufficiently large n, and any graph H of order , either H or its complement contains a (d,n)-common graph, that is, a graph in which every set of d vertices has at least n common neighbours.

Turán Numbers of Bipartite Graphs and Related Ramsey-Type Questions

TLDR
It is proved that, for any fixed bipartite graph H in which all degrees in one colour class are at most r, the Turán number is the maximum possible number of edges in a simple graph on n vertices that contains no copy of H.

On two problems in graph Ramsey theory

TLDR
This work improves the upper bound on the existence of a constant c such that, for any graph H on n vertices, rind(H) ≤ 2cnlogn, and moves a step closer to proving this conjecture.

Two remarks on the Burr-Erdos conjecture

ON THE MAGNITUDE OF GENERALIZED RAMSEY NUMBERS FOR GRAPHS

If G and H are graphs (which will mean finite, with no loops or parallel lines), define the Ramsey number r(G, H) to be the least number p such that if the lines of the complete graph Kp are colored

On graphs with linear Ramsey numbers

TLDR
In this paper, the use of the regularity lemma is avoided altogether, and it is shown that one can in fact take, for some ®xed c, c… † < 2 (log )2 in the general case, and even even 1.

On Bipartite Graphs with Linear Ramsey Numbers

TLDR
It is shown that for all and there exists a bipartite graph with n vertices and maximum degree at most whose ramsey number is greater than for some absolute constant c>1.

Ramsey numbers for sparse graphs

Hypergraph Packing and Sparse Bipartite Ramsey Numbers

  • D. Conlon
  • Mathematics
    Combinatorics, Probability and Computing
  • 2009
We prove that there exists a constant c such that, for any integer Δ, the Ramsey number of a bipartite graph on n vertices with maximum degree Δ is less than 2cΔn. A probabilistic argument due to