The minimum size of complete caps in (ℤ/nℤ)2

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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 languageEnglish (US)
Pages (from-to)1-19
Number of pages19
JournalElectronic Journal of Combinatorics
Volume13
Issue number1 R
Publication statusPublished - Jul 28 2006

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All Science Journal Classification (ASJC) codes

  • Discrete Mathematics and Combinatorics

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