## Abstract

For two graphs G^{<} and H^{<} with linearly ordered vertex sets, the ordered Ramsey number r_{<}(G^{<},H^{<}) is the minimum N such that every red-blue coloring of the edges of the ordered complete graph on N vertices contains a red copy of G^{<} or a blue copy of H^{<}. For a positive integer n, a nested matching NM_{n}^{<} is the ordered graph on 2n vertices with edges {i,2n−i+1} for every i=1,…,n. We improve bounds on the ordered Ramsey numbers r_{<}(NM_{n}^{<},K_{3}^{<}) obtained by Rohatgi, we disprove his conjecture by showing 4n+1≤r_{<}(NM_{n}^{<},K_{3}^{<})≤(3+5)n+1 for every n≥6, and we determine the numbers r_{<}(NM_{n}^{<},K_{3}^{<}) exactly for n=4,5. As a corollary, this gives stronger lower bounds on the maximum chromatic number of k-queue graphs for every k≥3. We also prove r_{<}(NM_{m}^{<},K_{n}^{<})=Θ(mn) for arbitrary m and n. We expand the classical notion of Ramsey goodness to the ordered case and we attempt to characterize all connected ordered graphs that are n-good for every n∈N. In particular, we discover a new class of ordered trees that are n-good for every n∈N, extending all the previously known examples.

Original language | English |
---|---|

Article number | 113223 |

Journal | Discrete Mathematics |

Volume | 346 |

Issue number | 2 |

DOIs | |

State | Published - 1 Feb 2023 |

Externally published | Yes |

## Keywords

- Nested matching
- Ordered Ramsey number
- Ordered graph
- Ramsey goodness

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics