On graphs with small Ramsey numbers *

@article{Kostochka2001OnGW,
  title={On graphs with small Ramsey numbers *},
  author={Alexandr V. Kostochka and Vojtech R{\"o}dl},
  journal={Journal of Graph Theory},
  year={2001},
  volume={37}
}
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 d and each γ,0 < γ < 1, there exists k = k(d,γ) such that for every bipartite graph G = (W,U;E) with the maximum degree of vertices in W at most d and $|U|\leq |W|^{\gamma }$, $R(G)\leq k|W|$. This answers a question of Trotter. We give also a weaker bound on the Ramsey numbers of graphs whose set… 
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51 Citations
Highly Influential Citations
4
Background Citations
19
Methods Citations
13

51 Citations

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References

SHOWING 1-6 OF 6 REFERENCES
The Ramsey number of a graph with bounded maximum degree
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…
Graphs with Linearly Bounded Ramsey Numbers
TLDR
It is proved that for each p ≥ 1, there is a constant c (depending only on p) such that the Ramsey number r ( G, G ) ≤ cn for eachp -arrangeable graph G of order n.
Subdivided graphs have linear ramsey numbers
  • N. Alon
  • Mathematics
    J. Graph Theory
  • 1994
TLDR
It is shown that the Ramsey number of any graph with n vertices in which no two vertices of degree at least 3 are adjacent is at most 12n, which settles the problem of Burr and Erdos.
Arrangeability and clique subdivisionsThe Mathematics of Paul Erdo }s
  • Arrangeability and clique subdivisionsThe Mathematics of Paul Erdo }s
  • 1997
Erdo}s, On the magnitude of generalized Ramsey numbers for graphs, in: `̀ In®nite and ®nite sets'
  • Colloquia Mathematica Soc. Janos Bolyai,
  • 1975