### Abstract

We investigate the following Ramsey-type problem. Given a natural number k, determine the smallest integer rr(k) such that, if n is sufficiently large with respect to k, and S is any set of n points in general position in the plane, then all but at most rr(k) points of S can be partitioned into convex sets of sizes ≥ k. We provide estimates on rr(k) which are best possible if a classic conjecture of Erdos and Szekeres on the Ramsey number for convex sets is valid. We also prove that in several types of combinatorial structures, the corresponding 'Ramsey-remainder' rr(k) is equal to the off-diagonal Ramsey number r(k, k - 1) minus 1.

Original language | English |
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Pages (from-to) | 519-532 |

Number of pages | 14 |

Journal | European Journal of Combinatorics |

Volume | 17 |

Issue number | 6 |

DOIs | |

Publication status | Published - Aug 1996 |

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

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## Cite this

Erdos, P., Tuza, Z., & Valtr, P. (1996). Ramsey-remainder.

*European Journal of Combinatorics*,*17*(6), 519-532. https://doi.org/10.1006/eujc.1996.0045