## Abstract

The Ramsey number, r(G), of a graph G is the minimum integer N such that, in every 2-colouring of the edges of the complete graph K_{N} on N vertices, there is a monochromatic copy of G. In 1975, Burr and Erdos posed a problem on Ramsey numbers of d-degenerate graphs, i.e., graphs in which every subgraph has a vertex of degree at most d. They conjectured that for every d there exists a constant c(d) such that r(G) ≤ c(d)n for any d-degenerate graph G of order n. In this paper we prove that r(G) ≤n^{1+0(1)} for each such G. In fact, we show that, for every ε > 0, sufficiently large n, and any graph H of order n_{1+ε}, 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. It is easy to see that any (d, n)-common graph contains every d-degenerate graph G of order n. We further show that, for every constant C, there is an n and a graph H of order Cn such that neither H nor its complement contains a (2, n)-common graph.

Original language | English (US) |
---|---|

Pages (from-to) | 627-641 |

Number of pages | 15 |

Journal | Combinatorics Probability and Computing |

Volume | 12 |

Issue number | 5-6 SPEC. ISS. |

DOIs | |

State | Published - Sep 2003 |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics