### Abstract

Suppose the edges of the complete graph on the natural numbers are colored with 2 colors. We show that (1) there is a monochromatic infinite path whose vertex set has upper density ≥ 2 3, and (2) there is a monochromatic infinite path such that, for infinitely many n, the set {1,..., n} contains at least the first .21n vertices of the path. We also consider the analogous problems for colorings with 3 or more colors.

Original language | English |
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Pages (from-to) | 59-70 |

Number of pages | 12 |

Journal | Discrete Mathematics |

Volume | 113 |

Issue number | 1-3 |

DOIs | |

Publication status | Published - Apr 5 1993 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics

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## Cite this

Erdos, P., & Galvin, F. (1993). Monochromatic infinite paths.

*Discrete Mathematics*,*113*(1-3), 59-70. https://doi.org/10.1016/0012-365X(93)90508-Q