## Abstract

The Ramsey number r(H) of a graph H is the minimum integer n such that any two-coloring of the edges of the complete graph K_{n} contains a monochromatic copy of H. While this definition only asks for a single monochromatic copy of H, it is often the case that every two-edge-coloring of the complete graph on r(H) vertices contains many monochromatic copies of H. The minimum number of such copies over all two-colorings of K_{r(H)} will be referred to as the threshold Ramsey multiplicity of H. Addressing a problem of Harary and Prins, who were the first to systematically study this quantity, we show that there is a positive constant c such that the threshold Ramsey multiplicity of a path or an even cycle on k vertices is at least (ck)^{k}. This bound is tight up to the constant c. We prove a similar result for odd cycles in a companion paper.

Original language | English (US) |
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Article number | 103612 |

Journal | European Journal of Combinatorics |

Volume | 107 |

DOIs | |

State | Published - Jan 2023 |

## All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics