## Abstract

We introduce and study a variant of Ramsey numbers for edge-ordered graphs, that is, graphs with linearly ordered sets of edges. The edge-ordered Ramsey number (Image found) of an edge-ordered graph (Image found) is the minimum positive integer N such that there exists an edge-ordered complete graph (Image found) on N vertices such that every 2-coloring of the edges of (Image found) contains a monochromatic copy of (Image found) as an edge-ordered subgraph of (Image found). We prove that the edge-ordered Ramsey number (Image found) is finite for every edgeordered graph (Image found) and we obtain better estimates for special classes of edge-ordered graphs. In particular, we prove (Image found) for every bipartite edge-ordered graph G on n vertices. We also introduce a natural class of edge-orderings, called lexicographic edge-orderings, for which we can prove much better upper bounds on the corresponding edge-ordered Ramsey numbers.

Original language | English |
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Pages (from-to) | 409-414 |

Number of pages | 6 |

Journal | Acta Mathematica Universitatis Comenianae |

Volume | 88 |

Issue number | 3 |

State | Published - 2 Sep 2019 |

Externally published | Yes |

## ASJC Scopus subject areas

- General Mathematics