### Abstract

Let A and B be two finite sets of points in the plane in general position (neither of these sets contains three collinear points). We say that A lies deep below B if every point from A lies below every line determined by two points from B and every point from B lies above every line determined by two points from A. A point set P is decomposable if either |P|=1 or there is a partition P_{1}∪P_{2} of P into nonempty and decomposable sets such that P_{1} is to the left of P_{2} and P_{1} is deep below P_{2}. Extending a result of Nešetřil and Valtr, we show that for every decomposable point set Q and a positive integer k there is a finite set P of points in the plane in general position that satisfies the following Ramsey-type statement. For any partition C_{1}∪⋯∪C_{k} of the pairs of points from P (that is, of the edges of the complete graph on P), there is a subset Q^{′} of P with the same triple-orientations as Q such that all pairs of points from Q^{′} are in the same part C_{i}. We then use this result to show that for every k there is a point set P such that no function Γ that maps ordered pairs of distinct points from P to a set of size k can satisfy the following property: if Γ attains the same values on two ordered triples of points from P, then these triples have the same orientation. Intuitively, this implies that there cannot be such a function that is defined locally and determines the orientation of point triples.

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

Pages (from-to) | 77-83 |

Number of pages | 7 |

Journal | Electronic Notes in Discrete Mathematics |

Volume | 61 |

DOIs | |

Publication status | Published - 1 Aug 2017 |

Externally published | Yes |

### Fingerprint

### Keywords

- induced Ramsey theorem
- order type
- point set
- point-set predicate

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Electronic Notes in Discrete Mathematics*,

*61*, 77-83. https://doi.org/10.1016/j.endm.2017.06.023