### Abstract

We consider the numbers associated with Ramsey's theorem as it pertains to partitions of the pairs of elements of a set into two classes. Our purpose is to give a unified development of enumerative techniques which give sharp upper bounds on these numbers and to give constructive methods for partitions to determine lower bounds on these numbers. Explicit computations include the values of R(3, 6) and R(3, 7) among others. Our computational techniques yield the upper bound R(x, y)≤cy^{x-1}log log y/logy for x≥3.

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

Pages (from-to) | 125-175 |

Number of pages | 51 |

Journal | Journal of Combinatorial Theory |

Volume | 4 |

Issue number | 2 |

State | Published - Mar 1968 |

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### Cite this

*Journal of Combinatorial Theory*,

*4*(2), 125-175.

**Some graph theoretic results associated with Ramsey's theorem.** / Graver, Jack E; Yackel, James.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory*, vol. 4, no. 2, pp. 125-175.

}

TY - JOUR

T1 - Some graph theoretic results associated with Ramsey's theorem

AU - Graver, Jack E

AU - Yackel, James

PY - 1968/3

Y1 - 1968/3

N2 - We consider the numbers associated with Ramsey's theorem as it pertains to partitions of the pairs of elements of a set into two classes. Our purpose is to give a unified development of enumerative techniques which give sharp upper bounds on these numbers and to give constructive methods for partitions to determine lower bounds on these numbers. Explicit computations include the values of R(3, 6) and R(3, 7) among others. Our computational techniques yield the upper bound R(x, y)≤cyx-1log log y/logy for x≥3.

AB - We consider the numbers associated with Ramsey's theorem as it pertains to partitions of the pairs of elements of a set into two classes. Our purpose is to give a unified development of enumerative techniques which give sharp upper bounds on these numbers and to give constructive methods for partitions to determine lower bounds on these numbers. Explicit computations include the values of R(3, 6) and R(3, 7) among others. Our computational techniques yield the upper bound R(x, y)≤cyx-1log log y/logy for x≥3.

UR - http://www.scopus.com/inward/record.url?scp=0010778447&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0010778447&partnerID=8YFLogxK

M3 - Article

VL - 4

SP - 125

EP - 175

JO - Journal of Combinatorial Theory

JF - Journal of Combinatorial Theory

SN - 0021-9800

IS - 2

ER -