### Abstract

R. H. Schelp conjectured that if G is a graph with |V(G)| = R(P _{n}, P _{n}) such that δ(G) > 3|V(G)|/4, then in every 2-colouring of the edges of G there is a monochromatic P _{n}. In other words, the Ramsey number of a path does not change if the graph to be coloured is not complete but has large minimum degree. Here we prove Ramsey-type results that imply the conjecture in a weakened form, first replacing the path by a matching, showing that the star-matching-matching Ramsey number satisfying R(S _{n}, nK _{2}, nK _{2}) = 3n - 1. This extends R(nK _{2}, nK _{2}) = 3n - 1, an old result of Cockayne and Lorimer. Then we extend this further from matchings to connected matchings, and outline how this implies Schelp's conjecture in an asymptotic sense through a standard application of the Regularity Lemma. It is sad that we are unable to hear Dick Schelp's reaction to our work generated by his conjecture.

Original language | English |
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Pages (from-to) | 179-186 |

Number of pages | 8 |

Journal | Combinatorics Probability and Computing |

Volume | 21 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - Jan 1 2012 |

### ASJC Scopus subject areas

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics

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## Cite this

*Combinatorics Probability and Computing*,

*21*(1-2), 179-186. https://doi.org/10.1017/S0963548311000599