### Abstract

We determine the 2-color Ramsey number of a connected triangle matching c(nK_{3}) that is any connected graph containing n vertex disjoint triangles. We obtain that R(c(nK_{3}), c(nK_{3}) = 7n-2 somewhat larger than in the classical result of Burr, Erdős, and Spencer for a triangle matching, R(nK_{3}, nK_{3}) = 5n. The motivation is to determine the Ramsey number R(C^{2} _{n}, C^{2} _{n}) of the square of a cycle C^{2} _{n}. We apply our Ramsey result for connected triangle matchings to show that the Ramsey number of an “almost” square of a cycle C^{2, c} _{n} (a cycle of length n in which all but at most a constant number c of short diagonals are present) is asymptotic to 7n/3.

Original language | English |
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Pages (from-to) | 109-119 |

Number of pages | 11 |

Journal | Journal of Graph Theory |

Volume | 83 |

Issue number | 2 |

DOIs | |

Publication status | Published - Oct 1 2016 |

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### Keywords

- Ramsey
- triangle matching

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Journal of Graph Theory*,

*83*(2), 109-119. https://doi.org/10.1002/jgt.21913

**Ramsey Number of a Connected Triangle Matching.** / Gyárfás, A.; Sárközy, Gábor N.

Research output: Contribution to journal › Article

*Journal of Graph Theory*, vol. 83, no. 2, pp. 109-119. https://doi.org/10.1002/jgt.21913

}

TY - JOUR

T1 - Ramsey Number of a Connected Triangle Matching

AU - Gyárfás, A.

AU - Sárközy, Gábor N.

PY - 2016/10/1

Y1 - 2016/10/1

N2 - We determine the 2-color Ramsey number of a connected triangle matching c(nK3) that is any connected graph containing n vertex disjoint triangles. We obtain that R(c(nK3), c(nK3) = 7n-2 somewhat larger than in the classical result of Burr, Erdős, and Spencer for a triangle matching, R(nK3, nK3) = 5n. The motivation is to determine the Ramsey number R(C2 n, C2 n) of the square of a cycle C2 n. We apply our Ramsey result for connected triangle matchings to show that the Ramsey number of an “almost” square of a cycle C2, c n (a cycle of length n in which all but at most a constant number c of short diagonals are present) is asymptotic to 7n/3.

AB - We determine the 2-color Ramsey number of a connected triangle matching c(nK3) that is any connected graph containing n vertex disjoint triangles. We obtain that R(c(nK3), c(nK3) = 7n-2 somewhat larger than in the classical result of Burr, Erdős, and Spencer for a triangle matching, R(nK3, nK3) = 5n. The motivation is to determine the Ramsey number R(C2 n, C2 n) of the square of a cycle C2 n. We apply our Ramsey result for connected triangle matchings to show that the Ramsey number of an “almost” square of a cycle C2, c n (a cycle of length n in which all but at most a constant number c of short diagonals are present) is asymptotic to 7n/3.

KW - Ramsey

KW - triangle matching

UR - http://www.scopus.com/inward/record.url?scp=84981507185&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84981507185&partnerID=8YFLogxK

U2 - 10.1002/jgt.21913

DO - 10.1002/jgt.21913

M3 - Article

AN - SCOPUS:84981507185

VL - 83

SP - 109

EP - 119

JO - Journal of Graph Theory

JF - Journal of Graph Theory

SN - 0364-9024

IS - 2

ER -