Characterizing optimal point sets determining one distinct triangle

Brenner, Hazel N.; Depret-Guillaume, James S.; Palsson, Eyvindur A.; Stuckey, Robert

doi:10.2140/involve.2020.13.91

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Characterizing optimal point sets determining one distinct triangle

Hazel N. Brenner, James S. Depret-Guillaume, Eyvindur A. Palsson and Robert W. Stuckey

Vol. 13 (2020), No. 1, 91–98

DOI: 10.2140/involve.2020.13.91

Abstract

We determine the maximum number of points in $ℝ^{d}$ which form exactly $t$ distinct triangles, where we restrict ourselves to the case of $t = 1$ . We denote this quantity by $F_{d} (t)$ . It is known from the work of Epstein et al. (Integers 18 (2018), art. id. A16) that $F_{2} (1) = 4$ . Here we show somewhat surprisingly that $F_{3} (1) = 4$ and $F_{d} (1) = d + 1$ , whenever $d \geq 3$ , and characterize the optimal point configurations. This is an extension of a variant of the distinct distance problem put forward by Erdős and Fishburn (Discrete Math. 160:1-3 (1996), 115–125).

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Keywords

one-triangle problem, Erdős problem, optimal configurations, finite point configurations

Mathematical Subject Classification 2010

Primary: 52C10

Secondary: 52C35

Milestones

Received: 12 February 2019

Revised: 29 September 2019

Accepted: 11 November 2019

Published: 4 February 2020

Communicated by Kenneth S. Berenhaut

Authors

Hazel N. Brenner
Department of Mathematics Virginia Tech Blacksburg, VA United States

James S. Depret-Guillaume
Department of Mathematics Virginia Tech Blacksburg, VA United States

Eyvindur A. Palsson
Department of Mathematics Virginia Tech Blacksburg, VA United States

Robert W. Stuckey
Department of Mathematics Virginia Tech Blacksburg, VA United States	Department of Mathematical Sciences Kent State University Kent, OH United States

Vol. 13, No. 1, 2020