### 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 |
---|---|

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 2012 |

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### ASJC Scopus subject areas

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

### Cite this

*Combinatorics Probability and Computing*,

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

**Star versus two stripes ramsey numbers and a conjecture of schelp.** / Gyárfás, A.; Sárközy, Göbor N.

Research output: Contribution to journal › Article

*Combinatorics Probability and Computing*, vol. 21, no. 1-2, pp. 179-186. https://doi.org/10.1017/S0963548311000599

}

TY - JOUR

T1 - Star versus two stripes ramsey numbers and a conjecture of schelp

AU - Gyárfás, A.

AU - Sárközy, Göbor N.

PY - 2012/1

Y1 - 2012/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84859329932&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84859329932&partnerID=8YFLogxK

U2 - 10.1017/S0963548311000599

DO - 10.1017/S0963548311000599

M3 - Article

AN - SCOPUS:84859329932

VL - 21

SP - 179

EP - 186

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

SN - 0963-5483

IS - 1-2

ER -