### Abstract

For a finite family F of fixed graphs let R_{k}(F) be the smallest integer n for which every k-coloring of the edges of the complete graph K_{n} yields a monochromatic copy of some F∈F. We say that F is k-nice if for every graph G with χ(G)=R_{k}(F) and for every k-coloring of E(G) there exists a monochromatic copy of some F∈F. It is easy to see that if F contains no forest, then it is not k-nice for any k. It seems plausible to conjecture that a (weak) converse holds, namely, for any finite family of graphs F that contains at least one forest, and for all k≥k_{0}(F) (or at least for infinitely many values of k), F is k-nice. We prove several (modest) results in support of this conjecture, showing, in particular, that it holds for each of the three families consisting of two connected graphs with 3 edges each and observing that it holds for any family F containing a forest with at most 2 edges. We also study some related problems and disprove a conjecture by Aharoni et al. (2015) regarding the size of matchings in regular 3-partite 3-uniform hypergraphs.

Original language | English (US) |
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Pages (from-to) | 29-44 |

Number of pages | 16 |

Journal | European Journal of Combinatorics |

Volume | 72 |

DOIs | |

State | Published - Aug 2018 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics

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## Cite this

*European Journal of Combinatorics*,

*72*, 29-44. https://doi.org/10.1016/j.ejc.2018.04.007