### Abstract

A coloring of the edges of a graph is called a local k-coloring if every vertex is incident to edges of at most k distinct colors. For a given graph G, the local Ramsey number, r_{loc}^{k}(G), is the smallest integer n such that any local k-coloring of K_{n}, (the complete graph on n vertices), contains a monochromatic copy of G. The following conjecture of Gyárfás et al. is proved here: for each positive integer k there exists a constant c = c(k) such that r_{loc}^{k}(G) ≤ cr^{k}(G), for every connected grraph G (where r^{k}(G) is the usual Ramsey number for k colors). Possible generalizations for hypergraphs are considered.

Original language | English |
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Pages (from-to) | 67-73 |

Number of pages | 7 |

Journal | Graphs and Combinatorics |

Volume | 3 |

Issue number | 1 |

DOIs | |

Publication status | Published - Dec 1987 |

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### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Mathematics(all)

### Cite this

*Graphs and Combinatorics*,

*3*(1), 67-73. https://doi.org/10.1007/BF01788530

**Linear upper bounds for local Ramsey numbers.** / Truszczynski, Miroslaw; Tuza, Z.

Research output: Contribution to journal › Article

*Graphs and Combinatorics*, vol. 3, no. 1, pp. 67-73. https://doi.org/10.1007/BF01788530

}

TY - JOUR

T1 - Linear upper bounds for local Ramsey numbers

AU - Truszczynski, Miroslaw

AU - Tuza, Z.

PY - 1987/12

Y1 - 1987/12

N2 - A coloring of the edges of a graph is called a local k-coloring if every vertex is incident to edges of at most k distinct colors. For a given graph G, the local Ramsey number, rlock(G), is the smallest integer n such that any local k-coloring of Kn, (the complete graph on n vertices), contains a monochromatic copy of G. The following conjecture of Gyárfás et al. is proved here: for each positive integer k there exists a constant c = c(k) such that rlock(G) ≤ crk(G), for every connected grraph G (where rk(G) is the usual Ramsey number for k colors). Possible generalizations for hypergraphs are considered.

AB - A coloring of the edges of a graph is called a local k-coloring if every vertex is incident to edges of at most k distinct colors. For a given graph G, the local Ramsey number, rlock(G), is the smallest integer n such that any local k-coloring of Kn, (the complete graph on n vertices), contains a monochromatic copy of G. The following conjecture of Gyárfás et al. is proved here: for each positive integer k there exists a constant c = c(k) such that rlock(G) ≤ crk(G), for every connected grraph G (where rk(G) is the usual Ramsey number for k colors). Possible generalizations for hypergraphs are considered.

UR - http://www.scopus.com/inward/record.url?scp=0002084641&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002084641&partnerID=8YFLogxK

U2 - 10.1007/BF01788530

DO - 10.1007/BF01788530

M3 - Article

AN - SCOPUS:0002084641

VL - 3

SP - 67

EP - 73

JO - Graphs and Combinatorics

JF - Graphs and Combinatorics

SN - 0911-0119

IS - 1

ER -