## Abstract

Letn be an integer greater than one. Our main result, called the "Structure Theorem" is that a set that containsn-1 disjoint continua that are cut by a single line cannot be ann-point set, that is, a set that meets every line in preciselyn points. This theorem unifies and significantly improves upon a number of known theorems. The second part of the paper is devoted to several theorems that address the question when a set that meets every line in at mostn points can be extended to ann-point set. These theorems also highlight the sharpness of the Structure Theorem. © 2003 Hebrew University.

Original language | English |
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Pages (from-to) | 321-354 |

Journal | Israel Journal of Mathematics |

Volume | 137 |

DOIs | |

Publication status | Published - 2003 |