### Abstract

If G and H are graphs, define the Ramsey number r(G, H) to be the least number p such that if the edges of the complete graph K_{p}are colored red and blue (say), either the red graph contains G as a subgraph or the blue graph contains H. Let mG denote the union of m disjoint copies of G. The following result is proved: Let G and H have k and l points respectively and have point independence numbers of i and j respectively. Then N - 1 ≤ r(mG, nH) ≤ N + C, where N = km + In - min (mi, mj) and where C is an effectively computable function of G and H. The method used permits exact evaluation of r(mGt nH) for various choices of G and H, especially when m — n or G-H. In particular, r(mK_{3}, nK_{3}) = 3m + 2n when m ≥ n, m ≥ 2.

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

Pages (from-to) | 87-99 |

Number of pages | 13 |

Journal | Transactions of the American Mathematical Society |

Volume | 209 |

DOIs | |

State | Published - 1975 |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Transactions of the American Mathematical Society*,

*209*, 87-99. https://doi.org/10.1090/S0002-9947-1975-0409255-0

**Ramsey theorems for multiple copies of graphs.** / Burr, S. A.; Erdös, P.; Spencer, Joel.

Research output: Contribution to journal › Article

*Transactions of the American Mathematical Society*, vol. 209, pp. 87-99. https://doi.org/10.1090/S0002-9947-1975-0409255-0

}

TY - JOUR

T1 - Ramsey theorems for multiple copies of graphs

AU - Burr, S. A.

AU - Erdös, P.

AU - Spencer, Joel

PY - 1975

Y1 - 1975

N2 - If G and H are graphs, define the Ramsey number r(G, H) to be the least number p such that if the edges of the complete graph Kpare colored red and blue (say), either the red graph contains G as a subgraph or the blue graph contains H. Let mG denote the union of m disjoint copies of G. The following result is proved: Let G and H have k and l points respectively and have point independence numbers of i and j respectively. Then N - 1 ≤ r(mG, nH) ≤ N + C, where N = km + In - min (mi, mj) and where C is an effectively computable function of G and H. The method used permits exact evaluation of r(mGt nH) for various choices of G and H, especially when m — n or G-H. In particular, r(mK3, nK3) = 3m + 2n when m ≥ n, m ≥ 2.

AB - If G and H are graphs, define the Ramsey number r(G, H) to be the least number p such that if the edges of the complete graph Kpare colored red and blue (say), either the red graph contains G as a subgraph or the blue graph contains H. Let mG denote the union of m disjoint copies of G. The following result is proved: Let G and H have k and l points respectively and have point independence numbers of i and j respectively. Then N - 1 ≤ r(mG, nH) ≤ N + C, where N = km + In - min (mi, mj) and where C is an effectively computable function of G and H. The method used permits exact evaluation of r(mGt nH) for various choices of G and H, especially when m — n or G-H. In particular, r(mK3, nK3) = 3m + 2n when m ≥ n, m ≥ 2.

UR - http://www.scopus.com/inward/record.url?scp=0003164032&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0003164032&partnerID=8YFLogxK

U2 - 10.1090/S0002-9947-1975-0409255-0

DO - 10.1090/S0002-9947-1975-0409255-0

M3 - Article

VL - 209

SP - 87

EP - 99

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

ER -