### Abstract

A typical problem arising in Ramsey graph theory is the following. For given graphs G and L, how few colors can be used to color the edges of G in order that no monochromatic subgraph isomorphic to L is formed? In this paper we investigate the opposite extreme. That is, we will require that in any subgraph of G isomorphic to L, all its edges have different colors. We call such a subgraph a totally multicolored copy of L. Of particular interest to us will be the determination of X_{s}(n, e, L), defined to be the minimum number of colors needed to edge‐color some graph G(n, ϵ) with n vertices and e edges so that all copies of L in it are totally multicolored. It turns out that some of these questions are surprisingly deep, and are intimately related, for example, to the well‐studied (but little understood) functions r_{k}(n), defined to be the size of the largest subset of {1, 2,…, n} containing no k‐term arithmetic progression, and g(n; k, l), defined to be the maximum number of triples which can be formed from {1, 2,…, n} so that no two triples share a common pair, and no k elements of {1, 2,…, n} span l triples.

Original language | English |
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Pages (from-to) | 263-282 |

Number of pages | 20 |

Journal | Journal of Graph Theory |

Volume | 13 |

Issue number | 3 |

DOIs | |

Publication status | Published - 1989 |

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

*Journal of Graph Theory*,

*13*(3), 263-282. https://doi.org/10.1002/jgt.3190130302