### Abstract

A natural generalization of graph Ramsey theory is the study of unavoidable sub-graphs in large colored graphs. In this paper, we find a minimal family of unavoidable graphs in two-edge-colored graphs. Namely, for a positive even integer k, let S_{k} be the family of two-edge-colored graphs on k vertices such that one of the colors forms either two disjoint K_{k / 2}'s or simply one K_{k / 2}. Bollobás conjectured that for all k and ε{lunate} > 0, there exists an n (k, ε{lunate}) such that if n ≥ n (k, ε{lunate}) then every two-edge-coloring of K_{n}, in which the density of each color is at least ε{lunate}, contains a member of this family. We solve this conjecture and present a series of results bounding n (k, ε{lunate}) for different ranges of ε{lunate}. In particular, if ε{lunate} is sufficiently close to frac(1, 2), the gap between our upper and lower bounds for n (k, ε{lunate}) is smaller than those for the classical Ramsey number R (k, k).

Original language | English |
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Pages (from-to) | 4396-4413 |

Number of pages | 18 |

Journal | Discrete Mathematics |

Volume | 308 |

Issue number | 19 |

DOIs | |

Publication status | Published - 6 Oct 2008 |

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### Keywords

- Ramsey theory
- Unavoidable structures

### Cite this

*Discrete Mathematics*,

*308*(19), 4396-4413. https://doi.org/10.1016/j.disc.2007.08.102