The no-three-in-line problem on a torus

2012/03/29 by Jim Fowler, Andrew Groot, Fowler, Jim +5 · 2 citations
Engineering · Mathematics · #05B30 #12Y05 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #graph theory and CDMA systems #math.AC #math.CO #msc:05B30 #msc:12Y05

paper · pdf · doi:10.48550/arxiv.1203.6604

10 pages, 3 figures

arxiv created 2012/03/29 · openalex publication_date 2012/03/29 · arxiv updated 2012/03/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let T(\Z_m × \Z_n) denote the maximal number of points that can be placed on an m × n discrete torus with "no three in a line," meaning no three in a coset of a cyclic subgroup of \Z_m × \Z_n. By proving upper bounds and providing explicit constructions, for distinct primes p and q, we show that T(\Z_p × \Z_p²) = 2p and T(\Z_p × \Z_pq) = p+1. Via Gröbner bases, we compute T(\Z_m × \Z_n) for 2 ≤ m ≤ 7 and 2 ≤ n ≤ 19.

Cited by

• A note on the no-three-in-line problem on a torus
• On No-Three-In-Line Problem on m-Dimensional Torus

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