## Abstract

A line in (ℤ/nℤ)^{2} is any translate of a cyclic subgroup of order n. A subset X ⊂ (ℤ/nℤ)^{2} is a cap if no three of its points are collinear, and X is complete if it is not properly contained in another cap. We determine bounds on Φ(n), the minimum size of a complete cap in (ℤ/nℤ)^{2}. The other natural extremal question of determining the maximum size of a cap in (ℤ/nℤ) ^{2} is considered in [8]. These questions are closely related to well-studied questions in finite affine and projective geometry. If p is the smallest prime divisor of n, we prove that max{4, √2p + 1/2 } ≤ Φ(n) ≤ max{4, p + 1}. We conclude the paper with a large number of open problems in this area.

Original language | English (US) |
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Pages (from-to) | 1-19 |

Number of pages | 19 |

Journal | Electronic Journal of Combinatorics |

Volume | 13 |

Issue number | 1 R |

State | Published - Jul 28 2006 |

## All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics