Ramsey numbers of degenerate graphs

@article{Lee2015RamseyNO,
  title={Ramsey numbers of degenerate graphs},
  author={Choongbum Lee},
  journal={arXiv: Combinatorics},
  year={2015}
}
  • Choongbum Lee
  • Published 18 May 2015
  • Mathematics
  • arXiv: Combinatorics
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 $H$ of chromatic number $r$ with $|V(H)| \ge 2^{d^22^{cr}}$ has Ramsey number at most $2^{d2^{cr}} |V(H)|$. This solves a conjecture of Burr and Erdős from 1973. 
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References

SHOWING 1-10 OF 42 REFERENCES
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.
On graphs with small Ramsey numbers *
TLDR
It is shown that for every positive integer d and each,0< <1, there exists kˆ k (d, ) such that forevery bipartite graph Gˆ (W,U;E ) with the maximum degree of vertices in W at most d and jU j j jW j, R (G ) k jWJ.
The Ramsey number of a graph with bounded maximum degree
A conjecture of Erdős on graph Ramsey numbers
On graphs with small Ramsey numbers
Let R(G) denote the minimum integer N such that for every bicoloring of the edges of KN, at least one of the monochromatic subgraphs contains G as a subgraph. We show that for every positive integer
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
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 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
A transference principle for Ramsey numbers of bounded degree graphs
We investigate Ramsey numbers of bounded degree graphs and provide an interpolation between known results on the Ramsey numbers of general bounded degree graphs and bounded degree graphs of small
...
...